Vectors and Linear Algebra?
I just started learning about linear algebra and vectors and there are some facts and practice questions I don’t quite get, it’s chapter one 
:
1) are all vectors orthogonal to another (not zero) vector in R^3 (3-D) constrained in a straight line or a plane? (having trouble visualizing)
2) if I let u = a unit vector, then P = proj u (v) would be (v*u)u ?
my * means dot product not multiplication
3) now I’m having trouble convincing myself with dot products
let c = any constant and then let x and y and z= vectors
can I do this all the time? (cx)*(cy) = c(x*y)
and would this always be true? (x*y)*z = x*(y*z)
and last question: if I had 2y – 3z = y*z where y and z are vectors, not variables, then is it actually possible to solve for y?
Thanks a tonne if someone can help me! even if it’s just a little!
One Response
Awms A
28 Feb 2010
For 2) yes, If u is a unit vector then proj u (v) = (v*u)u.
———
For 3)
(cx)*(cy) = c(x*y) is wrong, it should be
(cx)*(cy) = c^2 (x*y)
(x*y)*z makes no sense. x*y is a scalar, so we can’t take the dot product of a scalar and a vector.
Same problem with 2y – 3z = y*z. You have a vector on the left side and a scalar on the right side, so they can’t be equal. Since the equation itself makes no sense to have, then no, we can’t solve for y.