at 00:55 it should probably be noted that a sufficient condition for the orthogonality of two vector sums under consideration is that vectors of the same length must be summed: if A1 • B1 = 0, A2 • B2 = 0 and a1 = sqrt(A1•A1), a2 = sqrt(A2•A2),..., then |(A1 + A2) • (B1 + B2)| = |A1 • B2 + A2 • B1| = |cosφ * (a1*b2 - a2*b1)|, where cosφ = A1 • B2 / (a1*b2) = -A2 • B1 / (a2*b1). Therefore, in general, the vector sum (A1 + A2) is not always orthogonal to the vector sum (B1 + B2).
@@MathTheBeautiful Thank you for your reply and for sharing this great lecture series! Perhaps I misinterpret some of the statements and therefore think that they can lead to the wrong conclusion that one can check whether two vector sums (in 2D) are orthogonal to each other without summing the vectors, but only by knowing whether the corresponding terms are orthogonal (to be more clear, I was the one who made the wrong conclusion, and then, when I discovered that this method of checking for orthogonality did not work, I decided to make a post, although it is somewhat off topic of this video). (while I was writing all this, I also discovered that I don't know whether rotation is a linear transformation by definition or its linearity can be proven).
what happens with space curves in 3 dimensions? I'm trying to imagine it. is it M*ΔR where the rotation matrix M can be decomposed as M = (a 90 degree rotation)*(another rotation related to the torsion)?
at 00:55 it should probably be noted that a sufficient condition for the orthogonality of two vector sums under consideration is that vectors of the same length must be summed:
if A1 • B1 = 0, A2 • B2 = 0 and a1 = sqrt(A1•A1), a2 = sqrt(A2•A2),...,
then |(A1 + A2) • (B1 + B2)| = |A1 • B2 + A2 • B1| = |cosφ * (a1*b2 - a2*b1)|,
where cosφ = A1 • B2 / (a1*b2) = -A2 • B1 / (a2*b1).
Therefore, in general, the vector sum (A1 + A2) is not always orthogonal to the vector sum (B1 + B2).
That's a good point, but I don't believe I really relied on the inner product nature of orthogonality. I only used the linear property of rotation.
@@MathTheBeautiful Thank you for your reply and for sharing this great lecture series!
Perhaps I misinterpret some of the statements and therefore think that they can lead to the wrong conclusion that one can check whether two vector sums (in 2D) are orthogonal to each other without summing the vectors, but only by knowing whether the corresponding terms are orthogonal (to be more clear, I was the one who made the wrong conclusion, and then, when I discovered that this method of checking for orthogonality did not work, I decided to make a post, although it is somewhat off topic of this video).
(while I was writing all this, I also discovered that I don't know whether rotation is a linear transformation by definition or its linearity can be proven).
what happens with space curves in 3 dimensions? I'm trying to imagine it. is it M*ΔR where the rotation matrix M can be decomposed as M = (a 90 degree rotation)*(another rotation related to the torsion)?
Good question. The 3D case is different and more difficult, but yours truly cracked it! See this video: th-cam.com/video/tvVTzhq5dvo/w-d-xo.html