Answer to Question #1211 in Linear Algebra for huihui

Question

[b] u[/b] and [b]v[/b] are two non null vectors in [b]R[sub]n[/sub][/b],&
||u|| and ||v|| denote their respective lengths, and.&
||u – v|| denotes difference between u and v.
If ||u||=1 and ||v||=1 and ||u – v|| = ,& & then the u and v are&
& i. orthogonal&
& ii. Linearly independent&
& iii. Both i. and ii.&
& iv. All of them

Answer 1

As the length of the vector||u-v|| = sqrt (||u||2 + ||v||2 - 2 ||u|| ||v|| cos (180-α) ) = sqrt( 1 + 1 +2 cos (α) ) = √ 2;

2 = 2 + 2 cos(α);
cos(α) = 0;
α = 90;
The vectors are orthogonal and then linear independent.