Prove that if norm(u X v)=0 then u.v=a norm(v)^2 where a is any constant.
I know that both u and v are parallel vectors and that norm(u X v) sin x = 0
Then I got stuck.
Anyone knows any good resources for me to learn linear algebra (preferably from the basics, with exercises and answers)? I'm finding it v hard to understand this topic.
Last edited by renjer (2006-05-29 03:40:21)
Do you mean:
If the normal of u cross v is 0, then u dot v equals a times the normal of v squared.
Is that right?
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
Let me rephrase that, I made some mistakes in my original post.
Since the vectors are parrallel, cos θ = 1. So:
u.v = |u|*|v|
And thus, a = |u| / |v| so:
The key is to prove the two definations of dot product are equivalent.
Def1: dot product is the sum of corresponding cartesian co-ordinates.
Def2: dot product is the product of two vector lenths, and the cosine of their angle.
Usually this is proved within a triangle consisting of vector OA, OB, and AB=OB-OA
In a catesian system, distance formula is true based on Pythagoras' Theorem.
So if you define A(x[sub]1[/sub],y[sub]1[/sub],z[sub]1[/sub]) B(x[sub]2[/sub] ,y[sub]2[/sub],z[sub]2[/sub]), you get
meanwhile, Cosine Theorem
then below must be valid
Last edited by George,Y (2006-05-30 19:54:41)
I assumed that you can just assume that. Good proof George.