Find a nonzero vector in C³ orthogonal to two given vectors

by RoRi

April 28, 2016

Let

$A = (2,1,-i), \qquad B = (i,-1,2i).$

Find a non-zero vector $C$ in $\mathbb{C}^3$ orthogonal to both $A$ and $B$ .

For $C = (c_1, c_2, c_3)$ to be orthogonal to both $A$ and $B$ we must have $C \cdot A = 0$ and $C \cdot B = 0$ . This gives us

$\begin{align*} C \cdot A &= 0 & \implies && 2c_1 + c_2 + ic_3 &= 0 \\ C \cdot B &= 0 & \implies && -ic_1 - c_2 - 2ic_3 &= 0. \end{align*}$

From the first equation we get $c_2 = -2c_1 - ic_3$ . Plugging this into the second equation we then have $c_3 = (-2i-1)c_1$ . Let $c_1 = 1+i$ , then we have

$c_3 = 1-3i, \qquad c_2 = -5-3i.$

Therefore, $C = (1+i, -5-3i, 1-3i)$ is a vector in $\mathbb{C}^3$ orthogonal to both $A$ and $B$ .

Stumbling Robot

Find a nonzero vector in C³ orthogonal to two given vectors

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