5 ms·

Can anyone grow why in the first 5 minutes of part 1 he shows a 3 by 3 matrix multiples by a 1 by 3 vector yet verbally he pulls out of no where this idea that

by vertak 6y ago

Can anyone grow why in the first 5 minutes of part 1 he shows a 3 by 3 matrix multiples by a 1 by 3 vector yet verbally he pulls out of no where this idea that if you have _two_ 1 by 3 vectors that pass through the origin then their linear combinations can be represented by a plane? The jump from the 3D to the 2D has me lost and I gave up

swiley 6y ago
I don’t know which video you’re talking about but two non parallel vectors are enough to represent a plane, the normal will be their cross product. Also, if you have 3 dimensional vectors you were always in 3D.
neutronicus 6y ago
If you multiply the 1x3 vector by all scalars from -infinity to infinity you get all the points on a line. If you do the same for another 1x3 vector, and it is not parallel to the first, you get all the points on a different line. These two lines define a plane (and the cross product of the two vectors defines its normal vector)
jcmoyer 6y ago
The same concept applies in 2D, which might help you build the intuition to understand it in 3D. If you have a vector v=(1,0) that points to the right, you can scale this vector infinitely in that direction by multiplying it by a positive scalar. 5v = (5,0) 62.1v = (62.1,0) Similarly, you can scale that vector infinitely in the opposite direction (i.e. left) by multiplying it by a negative scalar: -987v = (-987,0) If we call this scalar c, the expression cv allows us to represent any point along the X axis simply by varying c, meaning that cv defines a line along that axis. Similarly, we can do the same for a vector w=(0,1) along the Y axis, scaling it by d. Now we have a method for moving to any point on the XY plane simply by varying c and d in the linear combination: cv + dw, meaning that we've defined a plane using two vectors. Two caveats: - this won't work if v and w are parallel; for example, if v = -w (and neither are zero) then we can only move along a line instead of a plane - it also won't work if either of the vectors are zero, because no matter what you multiply by, a zero vector can only represent a single point

by vertak 6y ago

swiley 6y ago

I don’t know which video you’re talking about but two non parallel vectors are enough to represent a plane, the normal will be their cross product. Also, if you have 3 dimensional vectors you were always in 3D.

neutronicus 6y ago

If you multiply the 1x3 vector by all scalars from -infinity to infinity you get all the points on a line. If you do the same for another 1x3 vector, and it is not parallel to the first, you get all the points on a different line. These two lines define a plane (and the cross product of the two vectors defines its normal vector)

jcmoyer 6y ago

The same concept applies in 2D, which might help you build the intuition to understand it in 3D. If you have a vector v=(1,0) that points to the right, you can scale this vector infinitely in that direction by multiplying it by a positive scalar. 5v = (5,0) 62.1v = (62.1,0) Similarly, you can scale that vector infinitely in the opposite direction (i.e. left) by multiplying it by a negative scalar: -987v = (-987,0) If we call this scalar c, the expression cv allows us to represent any point along the X axis simply by varying c, meaning that cv defines a line along that axis. Similarly, we can do the same for a vector w=(0,1) along the Y axis, scaling it by d. Now we have a method for moving to any point on the XY plane simply by varying c and d in the linear combination: cv + dw, meaning that we've defined a plane using two vectors. Two caveats: - this won't work if v and w are parallel; for example, if v = -w (and neither are zero) then we can only move along a line instead of a plane - it also won't work if either of the vectors are zero, because no matter what you multiply by, a zero vector can only represent a single point