[vertak]

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

[jcmoyer]

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

[neutronicus]

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)

[swiley]

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.