How do you find a unit vectors are orthogonal to both i+j and i+k?

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Answer 1 · 2016-11-19T14:26:55

$+-(-1/sqrt3, 1/sqrt3, 1/sqrt3)$

$a=i+j=<1, 1, 0) and b=i+k=<1, 0, 1>$

Let $c=+<(cos alpha, cos beta, cos gamma)>$ be the unit vectors (in

opposite directions) orthogonal to $a and b$ .

Then the scalar product $c.a = cos alpha + cos beta = 0$ .

Similarly, $c.b=cos alpha+cos gamma = 0$ .

It follows that $c = +- < -cos alpha, cos alpha, cos alpha>$ .

The directions are equally inclined to the axes, in the respective

octant ( the 2nd OX'YZ and 8th OXY'Z') , and so,

$cos alpha = +-1/sqrt3$

The answer is $+-(-1/sqrt3, 1/sqrt3, 1/sqrt3)$