find a unit vector with positive first coordinate orthogonal to both a & b!?

Question

If a= i + 3j + k, and b= i + 10j + k. What is i, j,k?

Answer 1

i, j, k are the standand unit vectors: i = (1,0,0), j = (0,1,0), and k = (0,0,1). You are looking for a vector xi + yj + zj such that the dot product with both a and b is 0. Thus, x + 3y + k = 0 (this is the dot product of our unknown vector with a) and x + 10 y + z = 0. By subtracting the first equation from the second, we see that 3y = -10y, so y = 0. Then both equations say that x + z = 0, so z = -x,

In addition, we want x to be positive and the unknown vector should be a unit vector. There are infinitely many ways to satisfy these requirements. For example, 1/sqrt(2)i + 0j - 1/sqrt(2)k is one such.