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Vector Algebra

Posted February 18th, 2011 by alpha
in

If a, b,c are three vectors of equal magnitude and all three of them are perpendicular to each other, then prove that they are(each of them are) inclined at an equal angle with the vector (a+b+c).

Seeking for an urgent help.Thanking You.

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Comments

scalar product

On February 19th, 2011 Structure says:

Scalar product

$$<u,v>=|u||v|\cos(u,v)$$

You have $ <a,a>=<b,b>=<c,c>=|a|^2 $ and $ <a,b>=<a,c>=<b,c>=0 $
You also have for $ \cos\alpha=\cos (a+b+c,a) $,$ \cos\beta=\cos (a+b+c,b) $,$ \cos\gamma=\cos (a+b+c,c), $

$$<a+b+c,a>=|a+b+c||a|\cos\alpha=<a,a>+<a,b>+<a,c>=|a|^2$$

So

$$\cos\alpha=\frac{|a|}{\sqrt{|a|^2+|b|^2+|c|^2}}=\frac{1}{\sqrt{3}}$$
$$\cos\beta=\frac{|b|}{\sqrt{|a|^2+|b|^2+|c|^2}}=\frac{1}{\sqrt{3}}$$
$$\cos\gamma=\frac{|c|}{\sqrt{|a|^2+|b|^2+|c|^2}}=\frac{1}{\sqrt{3}}$$

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