‹ Back to the paper
Solve the following.
Consider the two vectors and . Find the unit vector perpendicular to both and if .
Consider the two vectors $\vec{a} = 2\hat{i} - 3\hat{j} + 4\hat{k}$ and $\vec{b} = 5\hat{i} + q\hat{j} - \hat{k}$.
Find the unit vector perpendicular to both $\vec{a}$ and $\vec{b}$ if $q = 4$.
Answer
Answer
AIFor $q = 4$, $\vec{b} = 5\hat{i} + 4\hat{j} - \hat{k}$.
$\vec{a}\times\vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & -3 & 4 \\ 5 & 4 & -1 \end{vmatrix} = -13\hat{i} + 22\hat{j} + 23\hat{k}$, and $|\vec{a}\times\vec{b}| = \sqrt{169 + 484 + 529} = \sqrt{1182}$.
Unit vector perpendicular to both: $\pm\dfrac{-13\hat{i} + 22\hat{j} + 23\hat{k}}{\sqrt{1182}}$.
Final answer: $\pm\dfrac{-13\hat{i} + 22\hat{j} + 23\hat{k}}{\sqrt{1182}}$
From ISC 2025 Improvement Mathematics Paper 1, question 17(ii).
Check your working with the 3D geometry and vectors calculator: points, vectors, lines and planes: distances, angles, foot and image, shortest distance, intersections.