It works because of operator precedence: cross products comes first, dot products second. It's just like in ordinary arithmetic if someone asks you: "what is 'A plus B times C'?" The operational precedence forces parentheses around 'B times C', which must be the first stage even though it doesn't 'appear' first.
Should that be the cross dot product and not the dot cross product? Because cross dot product would be vector cross vector = vector and then dot vector, but dot cross product is vector dot vector = scalar cross vector, and that would be nonsense. Wouldnt it?
What do you mean by "linear combination" in case 3 and 4? Does it still apply if in a different exercise I would use subtraction? E.g. Plane 3 = Plane 2 - Plane 1 ? Or can it only be a sum of the two other ones?
How do we determine if there are 3 pairs of line of intersection (Shown in case 3 of this video) or its just 1 line of intersection (shown in case 4 of this video) ?
+Anurag Tiwari If the 3 planes (2 of them at a time) intersect in 3 parallel lines, then one of the normal vectors can be written as a linear combination of the other 2 normal vectors. That's not always easy to see by just looking at them.
+Anurag Tiwari I don't know of any easy test to determine if the 3 normals are dependent (1 is the linear combination of the other 2). Yes, the scalars can be negative numbers.
+Anurag Tiwari Yes, in fact the entire planes equations are dependent. There is a test that might help sometimes (I'm using *" to represent the dot product). If you call n1, n2 & n3 the three normals, then if n1*n2Xn3 is any non-zero value there must be a unique point of intersection. If n1*n2Xn3 = 0, then there may be a line of intersection or not solution at all.
It works because of operator precedence: cross products comes first, dot products second.
It's just like in ordinary arithmetic if someone asks you: "what is 'A plus B times C'?" The operational precedence forces parentheses around 'B times C', which must be the first stage even though it doesn't 'appear' first.
Very good. I learned useful knowledge. Thanls
Thanks very much, very clear explanation.
Super explanation.Understood consistency,inconsistency and number of solution to system of three variable equation
Very useful video, thanks!
excellent video!! thank you so much
Hlw
Should that be the cross dot product and not the dot cross product? Because cross dot product would be vector cross vector = vector and then dot vector, but dot cross product is vector dot vector = scalar cross vector, and that would be nonsense. Wouldnt it?
What do you mean by "linear combination" in case 3 and 4? Does it still apply if in a different exercise I would use subtraction? E.g. Plane 3 = Plane 2 - Plane 1 ? Or can it only be a sum of the two other ones?
It can be subtraction too.
god bless your soul good sir
How do we determine if there are 3 pairs of line of intersection (Shown in case 3 of this video) or its just 1 line of intersection (shown in case 4 of this video) ?
+Anurag Tiwari If the 3 planes (2 of them at a time) intersect in 3 parallel lines, then one of the normal vectors can be written as a linear combination of the other 2 normal vectors. That's not always easy to see by just looking at them.
+AlRichards314 Is that the only way we can determine that ? And in linear combination can we use negative numbers as the scalar multiple ?
+AlRichards314 If there is only one line of intersection (i.e. Case4); the normals will be a linear combination of other 2 normal vectors too right ?
+Anurag Tiwari I don't know of any easy test to determine if the 3 normals are dependent (1 is the linear combination of the other 2). Yes, the scalars can be negative numbers.
+Anurag Tiwari Yes, in fact the entire planes equations are dependent. There is a test that might help sometimes (I'm using *" to represent the dot product). If you call n1, n2 & n3 the three normals, then if n1*n2Xn3 is any non-zero value there must be a unique point of intersection. If n1*n2Xn3 = 0, then there may be a line of intersection or not solution at all.
Oh never mind! I wrote that before the end of the video, its just the way you wrote it. DURR
ly man