3 levels of "perpendicular"
HTML-код
- Опубликовано: 9 сен 2024
- Here's what I came up with for the difference between perpendicular, normal, and orthogonal. But at the end of the day, no one will get mad at you if you use one term over the other. The only exception might be "orthogonal" when you have to check the property by the dot production (aka inner product).
----------------------------------------
Big thanks to my Patrons for the full-marathon support!
Ben D, Grant S, Mark M, Phillippe S. Michael Z, Camille E.
Nolan C. Jan P. Devun C. Stefan C. Budoray Ethan BW
💪 Support this channel and get my math notes by becoming a patron: / blackpenredpen
🛍 Shop my math t-shirt & hoodies: amzn.to/3qBeuw6
----------------------------------------
#blackpenredpen #math #calculus #apcalculus
At the end of the day, perpendicular wins.
Thank you so much for your video
Sir can you pls solve the integral e^x/(sinx+cosx) I have tried a lot and want to solve it.
Plssssssss Plsssss sir 😢😢😢❤
Perpendicular vectors
Perpendicular matrix
Perpendicular projection
.
.
.
Well... All in all, we have to admit that orthogonal and normal are the Coolest😎🤣
For me orthogonal wins because i play with polynomials such as
Chebyshev, Legendre, Hermite,Laguerre
Orthogonalization is not the fastest way to get them
My take:
Perpendicular = Two straight intersecting objects.
Normal = Vector and Surface. Surface does not need to be flat, it just has to be smooth. Works on any number of dimensions for the surface, but the vector is a line.
Orthogonal = independent. Orthogonal vectors don't correlate. No amount of walking east will take me north. No amount of changing the value of a function at one point will change the value anywhere else. etc.
I tell my Precalculus students two lines can be perpendicular if they intersect at a right angle but in 3D space, the two lines may not intersect and we wouldn’t call them perpendicular in that case. But if you represent the lines as vectors then the vectors will be orthogonal.
Wacth this before you die:
ruclips.net/video/zOcFsuojgPc/видео.htmlfeature=shared😊
I never heard "normal" being used in math class, only hearing that for physics. I think it might be for the best since it could be confused with normalization and might make the name of Gram-Schmidt orthonormalization more confusing.
Then what do you call perpendicular to tangent
yep, I think the normal vector is only 'normal' in the sense that it is the bivector of size 1, which can be represented as an orthogonal vector of length one specifically in 3D, and by analogy we think of it the same when viewing 2D as a 3D slice, but it breaks down majorly in more dimensions.
(phrased a little simpler: the 'size 1' is what 'normal' means, and the 'preprendicular' part comes from treating a bivector as a perpendicular vector in 3D only)
@@anghme28ang11in arbitrary dimensions we call it the the orthonormal subspace of the tangent space in the ambient space (and often, the last two parts are implicitly understood, so it's just the orthonormal space)
i havent taken any physics classes yet but the only time i heard it was in a calc 3 course
Normalization is making it normal?
It's instructive for "orthogonal" to learn how the word is used in other contexts. In those other contexts, it tends to mean things that are completely unrelated to each other. For example, you could say that one political spectrum is orthogonal to another political spectrum (say, free market vs. protectionist, state religion vs. freedom of religion, wealth redistribution vs. entrenched aristocracy, authoritarian vs. libertarian - I'm not intending to say any of these are actually orthogonal, but just used as an example of things that might be argued to be orthogonal), in that where you stand on one spectrum has no bearing on where you stand on the other. Thus, you can think of them like linearly independent vectors that can form a basis of the space of political positions defined by those spectra. I used "orthogonal" this way for something related to work with my boss, and I had to explain what it meant, so it's not like every English speaker is familiar with its use in this way, but it does get used by academics.
unit normal vectors of a plane
so normal is not only for curves
I like to think of them as perpendicular being of things with the same dimensionality, i.e. two lines or two planes are perpendicular.
I think of normal as applying to things of different dimensions, like a line intersecting a plane. The line is normal if it is perpendicular to all lines in the plane that pass through that point.
Orthogonality is about independence in a more abstract sense, like how no amount of walking east will move you north. The area of a parallelogram is the product of a side with the orthogonal thickness. A change in one of the measurements has no change in the other measurement.
Ask yourself this question: Are the sine and cosine functions perpendicular, normal, or orthogonal to each other?
@@skilz8098orthogonal i think
@@cubing7276 Yup, but they are also all of the above!
@@skilz8098 in what sense are they perpendicular? Or normal?
@@divisix024 In the general base case coming straight from their definitions, not in a transformed case or other variations.
Normal comes from Latin "norma" which is a tool with a right angle, a carpenter's square.
Perpendicular comes from Latin perpendiculum "plumb line" and Late Latin perpendicularis "vertical, as a plumb line". Basically, it wants to say that the gravitational force is normal to the earth surface.
Orthogonal comes from grrek "orthos" (right/correct) and "gonia" (angle/corner).
It looks like math evolved from construction and physics (norma, perpendiculum) and became more abstract during the greek era turning into geometry and algebra.
For me it looks like the greek had some really smart people at this time leading the development of math. Their thinking and language looks a bit more abstract to me.
In some contexts, perpendicular means vertical (yet another word).
'Kathetal' is another word with a similar meaning.
All of these words roughy mean 'at right angles' which I believe is the literal meaning of the word orthogonal in Greek.
We are spoiled for choice in English, with its rich vocabulary, in many languages there aren't that many words to describe similar/nuanced situations
The terms Perpendicular, Normal and Orthogonal are NOT special terms of the English language. Pretty much any language has such terms.
Also, your "roughy mean 'at right angles'" is NOT accurate. The etymology of Perpendicular basically means "plumb line". This is NOT "at right angles", not even roughly.
The part
"... in many languages there aren't that many words to describe similar/nuanced situations"
is just laughable. Haha.
@@samueldeandrade8535 the plumb line was used to construct lines at right angle to the ground. All these terms basically mean the same thing, but in mathematics we use specific terms for specific scenarios for the sake of rigorous terminology.
In greek we just use kathetos and orthogonios, we dont have as many other terms
1. Another distinction is that the zero vector will be orthogonal to any other vector. This is important, for example, when you want the orthogonal complement of a set of vectors to be a subspace.
2. You said that the inner product is the same as the dot product for vectors in R^n. This is not quite true since there are non-standard inner products possible.
3. I think the term "normal" in mathematics is the champion for having the most different meanings.
There are normal distributions, normal subspaces, normalizing vectors, ...
Of course it is possible to quibble about what constitutes a "different"meaning...for example, adding integers could be seen as different from adding fractions.
Wacth this before you die:
ruclips.net/video/zOcFsuojgPc/видео.htmlfeature=shared😊❤❤
I think the normal vector is only 'normal' in the sense that it is the bivector (derived from the tangent plane) of size 1 (because it represents orientation without magnitude), which can be represented as an orthogonal vector of length 1 specifically in 3D, and by analogy we think of it the same when viewing 2D as a 3D slice, but it breaks down majorly in more dimensions.
Specifically what's 'normal' about it, is that it's size 1 (normalisation) not the perpendicular aspect (orthoganalisation), and we have it hammered into us in linear algebra that orthonormalisation comprises both normalisation and orthoganisation which are both independent things!
Personally, I have no issue with all three of them being interchangeable. Why? They all represent a translation of 90 degrees or PI/2 radians either it being a translation or a rotation. Consider the following if we take the value 1 which we typically perceive it as being a scalar and represent it by the vector And now let's add its additive inverse to it twice. + + which gives us the vector . This is the same exact thing as the basic arithmetic expression 1 + (-2) = -1 or simply 1 - 2 = -1. This two-step horizontal translation is the same thing as rotating the point about the fixed-point origin by 180 degrees in standard form (CCW).
We can introduce a new operator where I'll use .rot( ) to represent rotate by which can be in either degrees or radians.
We can then see that 1 - 2 is equivalent to 1.rot(180) or 1.rot(PI). Now, what happens to the value of 1 when we rotate it by 90 degrees or PI/2 radians?
We can easily visualize the expressions: 1.rot(90) and 1.rot(PI/2), yet there is no value along the "real number or horizontal x-axis" that can represent this value. If we go back to the vector notation of this scalar value We can then easily see that:
1.rot(90) and 1.rot(PI/2) gives the value or unit vector
What is so special about this?
By understanding the properties of the trig functions specifically the sine and cosine functions we know that they have the same range, domain, period, and wave form or shape, that they are continuous for all points along their curves, that their limits exist for all points. We know that they are periodic sinusoidal, rotational, circular, transcendental wave functions. They are for most tense and purposes the same function with slight differences. There are three main differences between them. The sine is an odd function, the cosine is an even function. They are literally 90-degree horizontal translations of each other. These first two are due to the third difference in their properties and that is their initial or starting positions. The sine starts at and the cosine starts at .
This here easily shows us that and are perpendicular, orthogonal, normal to each other. Since is a unit vector that is all three to the origin then it also implies that is also that too through induction since is also a unit vector.
Now when we take any numerical value (real) and rotate it by 180 degrees which is the same as performing the two-step linear translation of -2, we are changing its sign or its direction. If it's positive, it'll become its negative and if it's negative it will become its positive provided that the point of rotation remains fixed at the origin.
If we rotate by 360 degrees, then the value will remain the same. This goes to show that for all values, they are circular. 1, 2, 3, 1/2, sqrt(1), e, pi, etc. They are all circular.
Now when we rotate a value from the horizontal axis about the origin by 90 degrees, its displacement or translation becomes perpendicular or vertical to it. We can see this when we start to look at what are typically called imaginary numbers, but here I like to refer to them or call them the vertical numbers as they are perpendicular to the real or horizontal numbers. I tend to think of the Reals as being the Horizontal numbers and the Imaginaries as being the Vertical Numbers and that the composition of them still being the Complex Numbers. Why? When viewing it this way, it is easy to see that the Complex number a + bi is also the same as the vector from the origin to the point (a, bi) which is also the same as the point (cos(t) , i sin(t) ) in the complex plane. The cos part is the Horizontal (Real) or X proportion, and the i sin part is the Vertical (Imaginary) or Y proportion.
Not even from linear algebra itself but just from basic algebra and the formula to find a slope given two points along a line defined as rise/run as being the gradient that can be calculated by m = (y2-y1)/(x2-x1) = deltaY / deltaX is the same as finding sin(t)/cos(t) where t (theta) is the angle between the line y = mx+b where b = 0, and the +x-axis. This is the same as saying that m = deltaY/deltaX = sin(t)/cos(t) = tan(t). We can take the slope form of the line y = mx+b and substitute m with tan(t). Yes, there is a direct relationship between linearity and rotation as there is a direction relationship between linearity and trigonometric properties. This can be seen due to the fact that the Pythagorean Theorem: A^2 + B^2 = C^2, and the equation of a circle (X-h)^2 + (Y-k)^2 = r^2 where (X,Y) is a point on the circle, (h,k) is its center point and r is its radius are basically the same thing. The only major difference between them is that the Pythagorean Theorem is expressed in terms of Right Triangles and the Later is in terms of Circles.
Everything within mathematics can be reduced down and expressed in terms of addition, linear transformation, specifically translation. I've already demonstrated how a two-step horizontal translation in the opposite direction is equivalent to rotation and if we rotate it by only 90 degrees, we get its orthogonal counterpart which is the same as multiplying it by i. 3 * i = 3i which is the same as 3.rot(90) or 3.rot(PI/2). If we take the value 1 and rotate it by 45 degrees. 1.rot(45) or 1.rot(PI/4) we end up with the vector < sqrt(2)/2 , sqrt(2)/2)> as this point lies on the unit circle and if we are in the "real plane XY" it's simply (cos(t), sin(t)) and if we are in the Complex plane, it's simply (cos(t), i sin(t) ).
I do not see any difference between the three when it comes to being normal, orthogonal or perpendicular. However, within the context of normals while in the event of performing calculations on them, it is preferred to normalize your vectors so that their orthonormal components provide a unit vector. The reason for this typically within 3D Graphics and other fields as well such as within physics is that when it comes to rotations sometimes, we only care about its direction, so having a unit vector in that direction simplifies or reduces our expressions and computations. So, I can partially agree with that sentiment when considering normals; even within physics when calculating force vectors and having to find the normals, it's a similar thing but in a slightly different situation. Yet for most tense and purposes they all relate to being Right Angles or rotations by 90 degrees, or translations by PI/2 radians. This is why the sine and cosine functions as well as the other trig functions all have a Pythagorean Identity. This is also why the slope or gradient of a line, and the tangent functions are related and why we are even able to define derivatives and integrals based on the tangent (slope or gradient). This doesn't even include things such as the distance and midpoint formulas... Just some food for thought. And yet all of this can be derived from y = x.
Wacth this before you die:
ruclips.net/video/zOcFsuojgPc/видео.htmlfeature=shared😊
Normal means perpendicular, from latin, from norma which was the carpenters ruler that they used to make perpendicular lines. The later meaning of normal came later, from the sense of something made according to a ruler or a rule.
Great lead into the gyroscopic effect. Angular momentum is not causal or even real. It is a math analogy. The cause of the effect is found in the "perpendicular" tilt velocity created by tilting a spin plane. Opposite seesaw ends have opposite velocities, and spinning mass passing through these tilting ends of the plane must swap ends twice per rotation creating massive amounts of "perpendicular" acceleration focused at the tilting fulcrum axis running diametrically through the spin plane. All of this acceleration production is totally resolved in each instant, and it is perfectly "perpendicular" to spin velocity and the spin plane. Tilting is directional and is directed toward a tilting input initially, then it steers itself to align the fulcrum axis with the force causing the initial tilt input, and this is precession. The REASON this is a perpendicular effect is that the tilting fulcrum axis in the plane follows tilt direction like an axle being perpendicular to the direction of a car. Look at my video explanation of it called: Proof! Angular Momentum Is Not Causal.
Fantastic explanation. Undeerstood the difference today between perpendicular, normal and orthogonal after ages of doing math. Thanks so much bprp. As always, you are awesome.
Dot/inner product vanishing is only the first condition, the second condition is that neither of the vectors/functions are zero.
I would say that normal is a straight line in comparison to a line or surface (so the latter can be straight out curved)
So on my usage, perpendicular is a special case of normal.
In physics, light reflects off a mirror with the same angle compared too the normal: we use that for flat and curved mirrors alike.
"Normal" has other meanings. For instance, it is conjectured that all algebraic irrational numbers are normal in all bases.
Is there a name for that conjecture? I would like to look into it more.
N basis vectors for an N-dimensinal space don't need to be perpendicular, but they do need to be orthogonal. Different concepts. Orthogonal is a general concept beyond math or physics meaning changing one doesn't change the other; you can use it for any sort of things being measured to say that they aren't correlated.
basically, linearly independent
orthogonal and linearly independent are different things
you can have a basis that is not orthogonal (w.r.t. some metric/inner product)
@@MagicGonads A base of a vector space must be linear independent, but it doesn't have to be orthogonal. You can transform (I'm not sure atm if always) the base to an orthogonal base. Then if the vector space has a metric, you can even transform it to an orthonormal space, so the inner product of the base vectors is 0 if they are different ones or 1 if they are the same (e_i * e_j = 1 if I=j and 0 of i is not j)
But you don't need a metric to get an orthogonal Base
orthogonal means right-angled.
I've heard normal used for 2 straight things before, eg. a vector perpendicular to a plane in R^3 referred to as a normal vector of that plane
Perpendicular - both geometry can be 1d , 2d or 3d
Normal - one geometry have less dimension then other one , lets say one is line another is plane or curve
Orthogonal - both have same same dimensions
Также можно добавить Трансверсальность)
You can add transversal also
perpendicular = orthogonal (same thing used in different contexts)
normal = vector of size 1 (sometimes used with the implicit meaning of also being perpendicular, e.g. the unit vector of a plane and your example of the curve)
The inner product can also contain a weighting function. (f,g)=int_a^b f(x)g(x)w(x) dx For example, the Laguerre polynomials are orthogonal with a=0, b=inf, and w(x)=e^-x. Also, as a physicist, I probably wouldn't bat an eye if you used them interchangeably.
After u=sinx substitution, the limits should change to [-π,π] to [0,0], does not change the outcome though
BPRP! I NEED YOUR HELP NOW. PLEASE SOLVE [x^(sin(x)) = 1/e]. i've been messing with it for weeks and it seems like there is no way to solve it without a graph.
bunch of doing nothing left me with -csc(x)=ln(x)
no idea how to progress further
@@HellHeater why do I get the feeling lambert W or mclaurin/taylor series will be involved
Mathematica solves this?
Take log for on sides,
sin(x) ln(x) = -1
Now we are solving over real numbers:
As ln(x) is an increasing function for increasing x, and
-1
In summary:
1/ Perpendicular defined by the measurement of the angle between 2 objects (lines and planes). If the angle's value is 90 degrees, we use perpendicular.
2/ Normal defined through the tangent line or the tangent plane. It would involve the derivative and differential concepts. A normal line (or plane) of a curve (or a surface) is a line (or plane) perpendicular to the tangent line of that curve (or that surface).
3/ Orthogonal defined by the inner product. If the inner product of 2 objects is 0, we say they are orthogonal.
Is that correct?
Super class. I enjoyed it very much. Thank you so much 👍
Orthogonal is the Greek term and means
ορθος - right , γωνια - angle
at least according to Wikipedia
And what about ortho-normal!
I think of "normal" as a vector quantity with an orientation, whereas a "perpendicular" is a line without orientation.
Orthogonal is a generalization of perpendicular, but I usually think of it as "independent." Independence in statistics, having zero covariance, is the same as having a zero inner product.
Finally, for your integral, I would simplify (cos x)(cos 2x) = 1/2 (cos x + cos 3x), then a u-substitution becomes unnecessary.
what is a line without orientation? how you define orientation?
Spice it up with orthonormal, and linear independence. Nice video 😊
Would use the word normal in a physics class, when have a flat piece of glass, drawing normal to it to measure angle of incidence, reflection, refraction.
You should bring back the poke ball mic, I loved it!
They all the same, really.
Most of us, as normal human beings, would only encounter "good enough" spaces that are always locally homeomorphic to a Euclidean space.
No matter which convention or notation one is using, either it being Perpendicular, Normal, or Orthogonal, there is one common factor between all of them regardless of how they are being represented or used in any varying context and that is they are all symbolically related to being 90 degrees or PI/2 radians of separation. Another way to look at this from any of the three representations above in how they are typically used, is to also know that in all of those situations, the perpendicularity, the normalness, the orthogonality, etc where the dot product = 0, is the same thing as multiplying them by i or the sqrt(-1). The sqrt(-1) is a 90 degree rotation. This becomes more apparent when you start to work in various fields such as within physics typically within the field of electricity, magnetism, circuitry, etc... also within fields such as 3D Graphics Rendering with various transformations, translations, rotations, etc... where it's more efficient to use a 4D like object known as a quaternion to perform matrix to matrix, matrix to vector, matrix to quaternion, quaternion to vector calculations where quaternions themselves are a special case of the Hamiltonians. So, in truth, we can easily interchange the use of the words, Perpendicular, Normal, and Orthogonal to practically mean nearly the same thing. Sure, they are used to represent right angles in different contexts, but that's the one thing they all have in common, a factor of i.
In the example used in this video of _ cos(x) cos(2x) _ These two functions are not in phase, and so the angle between them is never a constant 90°
These two cosine functions are different _frequencies_ and distinctly different vectors in a frequency space.
It's like you if have a 7Hz and a 5Hz tone... the frequencies are not a combination or multiple of each other. (maybe I should use 50Hz and 60Hz or a pair of music pitches that sound bad together as an example)
Orthogonal is used in signal processing a lot. I don't quite grasp the significance of it though.
In "frequency space" the functions _cos(x)_ and _cos(2x)_ will be parallel lines... because they're different frequencies. (In a FFT spectral plot, it's two peaks on your graph. In amplitude time space, it's a squiggly cosine.)
No puede uno resistirse a decir: ¡Qué buen profesor!...
Those are just 2 examples of inner products, but it can be defined in many different ways for any vector space over R or C
If we assume that orthogonality interval is [-1;1]
then inner product presented on 6:43 works for Legendre polynomials because weight function is equal to 1
There are other ways to get Legendre polynomials
f.e
Polynomial solution of ODE
(1-x^2)y'' - 2xy' + n(n+1)y = 0
which satisfies condition y(1) = 1
By ordinary generating function
1/n!*d^n/dt^n (1/sqrt(1-2xt+t^2)) evaluated at t=0
Should inner product look as follows
(f,g) = \int\limits_{a}^{b}f(x)g(x)w(x)dx
where w(x) is weight function
In the Germany we use the words slightly different; in case i didn't mess up the translations, we say:
Perpendicularity is a concept of (classic) geometry; one (straight) line a is perpendicular to another (straight) line b, if and only if their intersections have 90° angles.
Orthogonality is a generalization of perpendicularity; in a vector space with a bilinear form , two vectors a and b are orthogonal, if and only if = 0.
The normal (/normal line) g (german: 'Normale' (/'normale Gerade')) of a hypersurface H (= straight/curved line, plane/surface, or higher dimensional generalization) is a straight line, whose direction vector n (we call normal vector) is orthogonal to the every tangent vector of H in the (specific) intersection point A (though g might intersect multiple times with H at different points under different angles).
We shorten that description by saying g and H are orthogonal in A (instead of mentioning the involved vectors) and also left out the 'in A' part in case it's obvious which point we mean, lke for example in case there is only one intersection point.
Thanks for the explanation. But when I was learning vector calculus, there was a term called 'Orthonormal'. What does that term mean?
Orthonormal
U missed the nucleus under the integral that makes two functions to be normal point to point. And on category 4 u could add convolution
The "Hesse normal form" describes a line or a plane by using a vector which is normal to it. That kind of contradicts what you said about the difference between "perpendicular" and "normal"...
Not if you extend "normal" to apply to lines or surfaces of any curvature (including straight/flat).
So perpendicular becomes a special case of normal, rather than something distinct.
Will you ever cover tensors or differential geometry !!? Great video btw
Hey, @blackpenredpen, speaking of orthogonality… do you have any content on Sturm-Liouville problems?
I also wondered about it during my Calc class
Résumé français.
1) Perpendiculaire= se """"croisent""" orthogonalement (ça nous semble évident pour des droites en 2D ou une droite et un plan/deux plans en 3D.
3) Orthogonal = définition de "l'angle droit" dépendante du nombre de dimensions dans lequel on travail, cette angle droit n'oblige pas forcément à ce que les objets se croisent pour de vrai - imaginer un angle droit à distance pour deux droites en deux dimensions.
2) Normal = orthogonal à la direction d'un objet, on peut imaginer un point de vue global ("normal à un plan" en terminale) ou un point de vue local comme le présente bbrp avec la normale à une courbe en 2d en un point mais bien entendu on peut étendre ces définition à des objets multidimensionnels ou inhabituels, c'est génial de parler de fonctions orthogonales !
Et surtout pour les post-bac, la définition d'orthogonal dépend de la définition du produit scalaire utilisé.
Merci BBRP♥ pour aborder ce thème pas facile à faire passer auprès des élèves.
if you knew what 'perpend' meant, you would argue this very differently.
modern mathematics terminology was mostly developed by people who were illiterate in Greek but didn't realize it, so they artificially changed what older terminology meant.
a casual of reading of Euclid reveals this to be abundantly accurate. where for instance Euclid specifically makes allowances for lines and surfaces to be curved, but working with such things is today called non-Euclidean, thanks to individuals like Gauss and Bolyai.
per - wall
pend - away
hence terms like 'perimeter' and 'pendulum'. so 'perpend' means for one thing to come off of another thing orthogonally.
consider related terms like:
- appendage: something hanging off a main body
- prepend: put before
- pericarp: outer lining of a seed
- pared: Spanish for 'wall'
- per: Coptic for 'house', later adopted into mathematical jargon to indicate division as containment. miles per hour is lit. miles housed in hours
- apeiron: boundless, lit. thing without walls, thing without boundaries
- perimeter: length of a wall, often misused in English now to mean 'wall' or outer boundary.
it's notable that in Euclid's Elements 'parallel' is never defined, but all of the foundational work on 'non-Euclidean' geometry is very confident that the meaning of 'parallel' which Euclid intended is a plainly established fact known to all.
in truth, 'parallel' never appears in the text outside of the phrase 'parallel straight lines'. what this means is that 'parallel' is a distinct property from straightness and not necessarily unique to lines. similarly, 'perpendicular' is also only used by Euclid with respect to straight lines. so the inference that these terms only apply to them is actually fallacious, and ignores the casual implications that Euclid was making with basic Greek vocabulary which he felt would be intuitive and obvious to his readers.
putting this all together, it should occur to you that the reason 'non-Euclidean' geometry was proposed 2 thousand years after Euclid is probably because earlier generations simply had a more correct understanding of what Euclid was intending to say. a clear example of this is the modern claim that Euclid's Elements introduces flatness and parallelism in his 5th postulate, despite the fact that he clearly outlines all of these principles in the preceding oroi. this is clear evidence that the modern interpretation of Euclid is rooted in simple illiteracy.
there are a number of conventional meanings, which are distinct both from each other and from the use implied
by Euclid. that is, ‘parallel lines’ are said to have one of the following properties:
- constant separation
- never intersecting
- same angles where crossed by some specific third line
- same angles where crossed by any third line
but note that none of these correspond to Euclid’s use.
- parallel objects are congruent objects positioned the same way relative to each other at all locations
- oros 23 specifically mentions parallel straight lines in the established context of a planed working surface
- straight lines are by definition mutually congruent
- parallel straight lines drawn upon a planed working surface never intersect as a consequence of the definition of
paralellism, not as an aspect of it. for example, parallel circles or squares can intersect
- lines in general are not congruent, as their curvature can differ in scale, degree and direction
- Euclid specifically says that the periphery of a circle is a line, which illustrates that he did not think of lines in
general as being straight, or potentially parallel to anything. later writers are far less careful
- because curved surfaces are necessarily objects embedded within a larger uncurved space, it is impossible to
generalize parallelism over them
- straight lines are not the only objects which can be parallel, for instance any two circles of the same diameter drawn upon
a planed working surface are parallel
-- such parallel circles may intersect, but never at corresponding points this is different from how great circles intersect on the surface of a sphere, as those intersections
-- specifically occur only at corresponding points
-- given that great circles are the only object which can be drawn on a spherical surface which can reasonably be said to be straight, parallel straight lines are impossible on a spherical surface since great circles cannot be translated over a spherical surface
-- similarly, the sorts of objects which might qualify as straight lines on a hyperbolic surface cannot be parallel, because their separation cannot be constant. and lines of constant separation on a hyperbolic surface cannot be straight. thus parallel straight lines are impossible on a hyperbolic surface because simple translation is not possible over a hyperbolic surface
what part of the modern interpretation exactly should I interpret as 'mathematical rigor'? without being explicitly coached to believe something very specific that doesn't follow from the sources, it's impossible to arrive at a conclusion which agrees with modern interpretations. this is a telltale sign of cult dogma.
your account of 'orthogonal vectors' dismisses the fact that the vectors you're comparing are by definition straight lines which share a boundary at the origin.
since there are two of them, we can find a plane which both of these lines lie in, and then if they are perpendicular within that plane, they are orthogonal. so your orthogonality test is testing for perpendicularity as you defined it anyway, you simply don't understand wtf you're talking about, so the fact that the method of computing orthogonality looks different to you has confused you into believing that it is somehow a fundamentally different beast altogether.
illiterate cult dogma.
I considered orthogonal for three mutually perpendicular lines or vectors
it's very informative and useful. Thanks a lot
Excellent, keep it up
When you made that sub U=sinx surely the limits change to 0,0 meaning you didn''t have to integrate
at all?
What will you say about normal form of line in that we have a perpendicular from origin to straight line?
Excellent explanation. 😃
Derivative prt 2 please 😅
Dear sir., will u solve this question without calculator..... 2^100??
Pependicular: zero inner product in real finite dimensional spaces.
Normal: perpendicular to a surface (surface = smooth d-1 dimensional manifold in a d-dimensional space).
orthogonal = zero inner product in any space (infinite dimensional, complex field, etc).
Tfw when you watch a math educator orthogonalize the words about orthogonalization 😂. Very meta! Also, have you considered “independent” from stats and maybe binormal/bitangent?
Need lvl 4 for OrthoNormal :D
Would you still say two lines (in 3-space) are perpendicular if they follow perpendicular directions but don't intersect?
Can you graph cos(x) and cos(2x) to demonstrate they are orthogonal?
In "frequency space" the functions _cos(x)_ and _cos(2x)_ will be parallel lines... because they're different frequencies. (In a FFT spectral plot, it's two peaks on your graph. In amplitude time space, it's a squiggly cosine.)
I'd like to see the difference in the plots of cos x and cos 2x over the range pi to -pi where they are orthogonal compared to a range where they are not orthogonal. What can we see in the function that causes their orthogonality?
Huh? The difference? cos 2x is just cos x squished. Also, what causes their orthogonality? Man, it seems you have lots to learn ...
@@samueldeandrade8535 So instead of being a condescending prick, you could provide some help?
These functions are two different _frequencies_ like the pitch of two musical notes that sound bad when played together.
In a "frequency space", these are parallel lines.
Please solve [x^(sin(x)) = 1/e] (without a graph)
I don’t think that equation can be solved algebraically!
7:15 This is a property of the dot product, true only when the coordinates have been defined through an orthonormal frame. That said, is the inner product dependent on the frame we are using in a certain space ?
In the _cos(x)_ and _cos(2x)_ example used, these are parallel lines in (Fourier transformed) frequency space... because they're different frequencies.
(In a FFT spectral plot, it's two peaks on your graph. In amplitude time space, it's a squiggly cosine.)
Now do binormal lines.
What did you do after highschool?
This is not your normal video.
It's mostly orthogonal 😁
This video is all right. 😅 Pun obviously intended.
Yes Sir❤
Now I completely understand the meaning of an "orthogonal height".
Hum? Height is, pretty much, orthogonal distance. Distance is something you can measure in some ways, it has more than one meaning.
What I meant is "orthogonal height" is kinda, KINDA, redundant.
@@samueldeandrade8535 Orthogonal heights are measured orthogonally from the gravity vector the level is set upon.
Lol...good question, I'm not sure...perhaps orthogonal can mean more, lol, like that the cross product in zero in contexts where perpendicularity in the geometric sense might not apply, etc, as happens in linear algebra and the like...but what is the difference between normal and perpendicular?...Lol...maybe the first can apply to any number of dimensions and the second to two dimensions?...Lol...I'm not sure I ever really asked myself this question, just accepting the new terms in different contexts, lol...
...and doesn't "normal" have another meaning?...Lol...the "normal"-part of the word "orthonormal" is referring to the norm, not perpendicularity, lol...
...as in "orthonormal basis"...at any rate, lol, I don't really know this one...
...Jesus, lol...I was trying to say inner product, not cross product...
A tangent to a circle is perpendicular to the radius and normal to the circle, then?
No, the normal to the circle is perpendicular to the tangent.
@@matthewparker9276 But the tangent is perpendicular to the radius. So it seems you're saying the radius might be normal to the tangent. Honestly not being deliberately obtuse, I'm actually confused.
@@Qermaq the normal is in the radial direction, yes.
... Calculus on my other channel...
👀😅🥲
why did u use -pi,pi and not any other number?
Because the period of the unmodified sine and cosine waves, is 2*pi radians. He could chose 0 to +2*pi, or -2*pi to 0, and get the same result, but he opted to chose -pi to +pi.
Then how does "orthocentre" get its name? I don't think "orthogonal" should not be used here because it is not related to vectors / functions etc.
Orthocenter gets its name because it is the point where three lines (the altitudes) that are orthogonal to the sides of the triangle, intersect. Should it really be called the perpencenter? Perhaps, but for historical reasons, it's called the orthocenter. Probably has to do with conventions for using Greek over Latin, when naming triangle terminology.
@@carultch I see. Thanks for the explanation.
Hmm. Reading different books by different authors about different math topics on different levels uses different definitions thus resulting in different comments to this video (and all others).
5:13 i have never seen anybody write vectors like that. can someone explain that
This is in a notation of row vector where the components are in a row. You can also write it in a column vector form with the components in a column.
its basically the coefficients of i, j, k and l which are the unit vectors in mutually perpendicular directions
If vector u is the origin to and vector v is the origin to the dot product is 0 so they are orthogonal, but they are not perpendicular, as they make a non-right angle. Do I have that right?
They do make a right angle. Draw it and see.
@@stephenbeck7222 Dammit I had the number backward in my head.
Now can you do a video that describes the difference between "mean" and "average". I have read they don't mean the the exact same thing.
The *mean* usually means the arithmetic mean (add your numbers, divide by how many you have). *Average* is fine in English but considered non-rigorous in math, as it can refer to different kinds of averages.
@@TheOiseau As you hinted, there are different kinds of mean too: arithmetic, geometric, harmonic and no doubt others. Non-mean averages include the mode and the median.
"mean" is the sum of the values divided by the number of values. "average" can refer to one of several different things, usually mean, median, or mode. Usually it refers to the mean.
@@vibaj16 Wikipedia "Pythagorean Means" article describes four different means and how they are related.
Perpendicular is to Normal as Parallel is to _______ ?
8th bprp go brrrrrrrr
If there is an inner product of 2 functions, does that mean that there is an outer product of 2 functions?
Uhum. Also called tensor product.
Black Pen Red Pen Blue Pen 🤔🤔
There's also orthonormal. But don't know how this applies.
(for 2 bases)
@@zirkq thanks.
@@arequina Orthonormal means a system of unit vectors to define a space for other vectors to exist, where the unit vectors are both all perpendicular to each other, and all have a magnitude equal to 1. It's more like orthounity, than orthonormal.
Perpendicular : kabhi letke hilaya hai ?😅
From Bharat
Never heard of "normal line" in my entire math history. I would call it a perpendicular line to the tangent at the tangent point. Sure, it is a mouthful, but it happens so rarely, that you don't need a special word in my opinion.
It depends on what convention people agreed on or wanted to use. In my experience with UK’s metric: “normal line,” is fairly and commonly used among mathematical analysis (in general as per its definition) and may extend to various applications such as Snell’s law. And funnily, people soon grew with it.
Overall, mathematics as a field - whether people wanted to generalise it as lexicon or a language, it’s up to them - is simply also about getting our points across as in: “How to objectively quantify this reality in a meaningful way or unit?” Or, “How to prove, express or evaluate the workings?” Et cetera.
We don't call it a "normal line". We just call it a "normal".
@@peterchan6082 A normal usually means a unit vector aligned with the perpendicular line. That's why it's called "normal", it's "normalized" so it has a length of 1.
Cool
👍💯
✨✨✨👨🎓👨🎓👨🎓👨🎓✨✨✨✨✨
In intro physics, students do many problems where they are required to draw a normal to the surface of a ramp.
The existence of almost synonyms is bothering bprp.
...
I can't see why. This makes no sense. Oh my Euler ... I just heard him saying Orthogonal is the most mathematical term compared to Perpendicular and Normal??? Wtf² ...
25th
57th
TL;DW; Basically nothing.
Can you solve this integration bug in your channel \int_{1}^{\infty} \left( \frac{\ln(x)}{x}
ight)^{2011} \, dx Thanks
This does not converge, try checking using integration by substitution.
Didn't Prime Newtons just post a video on this?
Second
^2
Reminded me of this old goodie: ruclips.net/video/BKorP55Aqvg/видео.html
Holding a lavalier microphone...
This is the first part
ruclips.net/video/QBJQtcA8lEg/видео.htmlsi=6EPdANKj8-p2xdQ_
wsg
Orthogonal = right angle from Greek roots. If a right angle isn't a straight line being perpendicular to another straight line in 2D, what is it?! It's useless to make up meaningless terms. Too "smart aleck-y" for their own good. There's no such thing as a "normal" (Is that 正常, 普通 or 一般 in Chinese, sir?) math problem either. 😂😅😊😢😮