# Proof of vectors

• Feb 25th 2007, 08:28 AM
faure72
Proof of vectors
this one problem on my practice sheet for my linear algebra test was confusing me, so i was wondering if anyone could show me how it is done in case i receive another problem like it...

Given vectors v1, v2, v3, suppose that v1 = xv2 for some scalar x. Prove that if v2 is perpendicular to v3, then v1 is perpendicular to v3.
• Feb 25th 2007, 08:43 AM
topsquark
Quote:

Originally Posted by faure72
this one problem on my practice sheet for my linear algebra test was confusing me, so i was wondering if anyone could show me how it is done in case i receive another problem like it...

Given vectors v1, v2, v3, suppose that v1 = xv2 for some scalar x. Prove that if v2 is perpendicular to v3, then v1 is perpendicular to v3.

Two vectors a and b are perpendicular if a (dot) b = 0

Now we know that v2 (dot) v3 = 0

So v1 (dot) v3 = [xv2] (dot) v3 = x[v2 (dot) v3] = x[0] = 0.

-Dan
• Feb 25th 2007, 08:48 AM
Soroban
Hello, faure72!

Two vectors u and v are perpendicular if and only if u·v = 0.

Quote:

Given: vectors v1, v2, v3.
Suppose that v1 = x(v2) for some scalar x.
Prove that if v2 is perpendicular to v3, then v1 is perpendicular to v3.

We are given: .v1 = x(v2) . . v2 = v1/x .[1]

And we are given: .v2 perp v3 . . v2·v3 = 0 .[2]

Substitute [1] into [2]: .(v1/x)·v3 .= .0 . . (v1·v3)/x .= .0 . . v1·v3 .= .0

Since v1·v3 = 0, then: .v1 perp v3.