How do you find a unit vector that is orthogonal to both u = (1, 0, 1) v = (0, 1, 1)?

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Answer 1 · 2016-07-23T07:15:20

$(u xx v) / (|| u xx v ||) = (-sqrt(3)/3, -sqrt(3)/3, sqrt(3)/3)$

The cross product of $u = (u_1, u_2, u_3)$ and $v = (v_1, v_2, v_3)$ is given by:

$(u_1, u_2, u_3) xx (v_1, v_2, v_3) = (abs((u_2, u_3),(v_2, v_3)), abs((u_3, u_1),(v_3, v_1)), abs((u_1, u_2),(v_1, v_2)))$

This will be orthogonal to both $u$ and $v$ , but will need scaling to make it unit length.

So we find:

$u xx v = (1, 0, 1) xx (0, 1, 1)$

$= (abs((0, 1),(1, 1)), abs((1, 1),(1, 0)), abs((1, 0),(0,1)))$

$= (-1, -1, 1)$

Then:
$||""(-1, -1, 1) || = sqrt((-1)^2+(-1)^2+1^2) = sqrt(1+1+1) = sqrt(3)$

So to make $(-1, -1, 1)$ into a unit vector, divide it by $sqrt(3)$ :

$1/sqrt(3) (-1, -1, 1) = sqrt(3)/3 (-1, -1, 1) = (-sqrt(3)/3, -sqrt(3)/3, sqrt(3)/3)$