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

Question

29038 views around the world

You can reuse this answer
Creative Commons License

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)#