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.
Hello, faure72!
Two vectors u and v are perpendicular if and only if u·v = 0.
We are given: .v1 = x(v2) . → . v2 = v1/x .[1]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.
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.