Question

Find a unit vector orthogonal to both \(u=<2,4,-1>\) and \(v=<0,-3,2>\).

Short answer

The unit vector is \(\frac{5}{\sqrt{77}}i-\frac{4}{\sqrt{77}}j-\frac{6}{\sqrt{77}}k \).

Step-by-step solution

Given Information

The given vectors are \(u=<2,4,-1>\) and \(v=<0,-3,2>\).

The orthogonal vector is \(\vec{b}=5i-4j-6k\).

The unit vector \(\hat{b}=\frac{\vec{b}}{|b|}\).

Find the unit vector

\(\begin{align*}\hat{b}&=\frac{5i-4j-6k}{\sqrt{5^2+(-4)^2+(-6)^2}}\\&=\frac{5}{\sqrt{77}}i-\frac{4}{\sqrt{77}}j-\frac{6}{\sqrt{77}}k\end{align*}\)