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

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Answer 1 · 2017-01-22T15:19:10

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 the dot product 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))#