# orthogonal vectors

• January 19th 2009, 11:50 AM
viet
orthogonal vectors
I don't understand how to solve this problem, help please.

Quote:

The Vectors $U = cos\theta I+sin\theta J, V = -sin\theta I+cos\theta J$ for any $\theta$, form an orthonormal basis for the plane; they are orthogonal vector of length 1. Let $\theta = \frac{2\pi}{4}$ and $x = 5I+8J$. Then we can write $X = uU + vV$

with $u$ = ___, $v$ = ___
• January 19th 2009, 12:20 PM
Jester
Quote:

Originally Posted by viet
I don't understand how to solve this problem, help please.

Substitute $\theta = \frac{2 \pi}{4} = \frac{\pi}{2}$ into both $U\; \text{and} \; V$ giving $U = \cos \frac{\pi}{2} i + \sin \frac{\pi}{2}\, j \; = j \;\;\text{and} \; V = - \sin \frac{\pi}{2}\, i + \cos \frac{\pi}{2} j = - i$. So $X = u U + v V = u j + v (-i) = 5 i + 8 j$. Now compare.