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Answer to Question #2167 in Linear Algebra 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:

The angles can be obtained in the following way:
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