Question

If u = 2i − 3j + k, v = i − 2k, w = ai + bj + ck form an orthogonal basis of R3 , find the 
possible values of a, b and c. 
Further, obtain the angles x = i − 3j− 3k makes with each of these vectors.  
b) Find the direction cosines of the perpendicular from the origin to the plane 
3r.(2i − 3j+ k) + 7 = 0.

Accepted Answer

Lets construct the system of equations for a,b,c. By the definition of
the orthonormal basis:

The angles can be obtained in the following way:
[https://latex.codecogs.com/gif.latex?%7C%5Cvec%7Bx%7D%7C%20=%20%5Csqrt%7B1+9+9%7D=%20%5Csqrt%7B19%7D%20%5Chspace%7B6mm%7D%20%7C%5Cvec%7Bu%7D%7C=%7C%5Cvec%7Bv%7D%7C=%7C%5Cvec%7Bw%7D%7C=1%20%5C%5C%20angle[x%20u]%20=%20%5Carccos%7B%28%5Cvec%7Bx%7D%5Ccdot%5Cvec%7Bu%7D%29/%7Cx%7C%7D%20=%20%5Carccos%20%7B%5Cfrac%7B8%7D%7B%5Csqrt%7B19%7D%7D%7D%5C%5C%20angle[x%20v]%20=%20%5Carccos%7B%28%5Cvec%7Bx%7D%5Ccdot%5Cvec%7Bv%7D%29/%7Cx%7C%7D%20=%20%5Carccos%20%7B-%5Cfrac%7B5%7D%7B%5Csqrt%7B19%7D%7D%7D%20%5C%5C%20angle[x%20w]%20=%20%5Carccos%7B%28%5Cvec%7Bx%7D%5Ccdot%5Cvec%7Bw%7D%29/%7Cx%7C%7D%20=%20%5Carccos%20%7B-%5Cfrac%7B18%7D%7B%5Csqrt%7B19%7D%5Csqrt%7B70%7D%7D%7D%20%5C%5C]
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