orthogonal vectors

**viet** · January 19th 2009, 11:50 AM

I don't understand how to solve this problem, help please.

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 $u$ = ___, $v$ = ___

**Danny** · January 19th 2009, 12:20 PM

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.

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