Question

Find the unit vector in the direction of \[3 \hat{i} + 4 \hat{j} - 12 \hat{k} .\]

Answer 1

Let \[\vec{a} = 3 \hat{i} + 4 \hat{j} - 12 \hat{k} .\]
Then,
\[\left| \vec{a} \right| = \sqrt{3^2 + 4^2 + \left( - 12 \right)^2} = \sqrt{9 + 16 + 144} = \sqrt{169} = 13\]
So, a unit vector in the direction of \[\vec{a}\] is given by

\[\hat{a} = \frac{\vec{a}}{\left| \vec{a} \right|} = \frac{1}{13} \left( 3 \hat{i} + 4 \hat{j} - 12 \hat{k} \right) = \frac{3}{13} \hat{i} + \frac{4}{13} \hat{j} - \frac{12}{13} \hat{k}\]