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Answer on Linear Algebra Question for biju

Question #2167
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.
Expert's answer
Let's construct the system of equations for a,b,c. By the definition of the orthonormal basis:

/cgi-bin/mimetex.cgi?%5Cvec%7Bu%7D%5Ccdot%20%5Cvec%7Bw%7D%20=%20%5Cvec%7Bv%7D%5Ccdot%5Cvec%7Bw%7D%5Cequiv%200%5C%5C%20%7C%5Cvec%7Bw%7D%7C=1.%5C%5C%20%5Cvec%7Bu%7D%20=[2,-3,1],%20%5C%20%5Cvec%7Bv%7D=[1,0,-2],%20%5C%20%5Cvec%7Bw%7D=[a,b,c]%20%5C%5C%20%5C%5C%20%5Cvec%7Bu%7D%5Ccdot%20%5Cvec%7Bw%7D%20=%202a%20-3b%20+%20c%20=0%20%5C%5C%20%5Cvec%7Bv%7D%5Ccdot%20%5Cvec%7Bw%7D%20=%20a%20-%202c%20=0%20%5C%5C%20a%5E2+b%5E2+c%5E2%20=%201%5C%5C%20%5C%5C%20a%20=%202c,%20%5C%20b%20=%20%5Cfrac%7B5%7D%7B3%7Dc,%20%5C%5C%20c%5E2%284%20+%20%5Cfrac%7B25%7D%7B9%7D%20+1%29%20=%201%20%5C%5C%20c%20=%203/%5Csqrt%7B70%7D,%20%5C%20a%20=%206/%5Csqrt%7B70%7D,%20%5C%20b%20=%205/%5Csqrt%7B70%7D

The angles can be obtained in the following way:
http://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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