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QUESTION

How do you find all unit vectors orthogonal to v=i+j+k?

A generalised unit vector is:

mathbf u = 1/(sqrt( 2 (alpha_2 ^2 + alpha_3 ^2 + alpha_2 alpha_3)))((- alpha_2 - alpha_3),(\alpha_2),(alpha_3))

There are an infinite number of vectors thare are orthogonal to mathbf v = ((1),(1),(1)).

If mathbf alpha is one such vector, we know from that mathbf v * mathbf alpha = 0 implies alpha_1 + alpha_2 + alpha_3 = 0

A generalised vector is therefore:

mathbf alpha = ((- alpha_2 - alpha_3),(\alpha_2),(alpha_3))

A generalised unit vector is:

mathbf u = 1/(sqrt( 2 (alpha_2 ^2 + alpha_3 ^2 + alpha_2 alpha_3)))((- alpha_2 - alpha_3),(\alpha_2),(alpha_3))