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Given two vectors, find a nonzero vector orthogonal to both

by RoRi

Let A = (1,-1,2) and B = (2,1,-1). Find a vector C \neq O that is orthogonal to both.

Since C = (c_1, c_2, c_3) must be orthogonal to both A and B we have

    \begin{align*} A \cdot C &= 0 & \implies && c_1 - c_2 + 2c_3 &= 0 \\ B \cdot C &= 0 & \implies && 2c_1 + c_2 - c_3 &= 0. \end{align*}

(As usual, if you know some matrix algebra there are easier ways to solve this. We’ll do it the hard way.) From the first equation we have c_1 = c_2 - 2c_3. This gives us an equation

    \[ 2c_1 + c_2 - c_3 = 0 \quad \implies \quad 2c_2 - 4c_3 + c_2 - c_3= 0 \quad \implies \quad c_2 = \frac{5}{3} c_3. \]

Since c_3 is then arbitrary we choose c_3 = -3 and obtain

    \[ c_2 = -5 \qquad c_1 = 1. \]

Therefore, C = (1,-5,-3).

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