Vector problem?

Question

How would you find two nonparallel vectors both orthogonal to <1,1,1>?

Answer 1

Vectors are orthogonal iff their dot product is zero. For instance if V1 = a*i + c*j + d*k (where i,j,k, are three dimensional orths) and V2 = x*i + y*j + z*k, then dot product V1.V2 = ax+by+cz. If we take your vector V1 = i + j + k, then V2 is orthogonal on your vector iff V1.V2 = x + y + z = 0. Therefore any vector whose x, y, z, components satisfy the above equation is orthogonal to your vector. On the other hand vectors are parallel if their components are scaled by the same multiplicative factor. For instance vector parallel with V2 is V3= s*(xi + yj + zk) where s is scalar constant. Therefore, two nonparallel vectors orthogonal to yours are V2 = (x2, y2, z2) with x2+y2+z2=0, and V3 = (x3,y3,z3) with x3+y3+z3=0 such that x2, y2, z2 are not proportional to x3, y3, z3. As you could imagine there are infinite number of such pairs. To pick up two: V2=(-1, -1, 2), V3=(2, -1, -1).