GATE 2022 ME (ME1) – Question 48

Engineering Mathematics · Calculus: Gradient, divergence, curl, vector identities, line, surface and volume integrals · 2 marks · Numerical answer

Consider two vectors: $\vec{a} = 5\mathbf{i} + 7\mathbf{j} + 2\mathbf{k}$, $\vec{b} = 3\mathbf{i} - \mathbf{j} + 6\mathbf{k}$
Magnitude of the component of $\vec{a}$ orthogonal to $\vec{b}$ in the plane containing the vectors $\vec{a}$ and $\vec{b}$ is __________ (round off to 2 decimal places).

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Correct answer: 8.28 to 8.36

Explanation

The component of $\vec{a}$ perpendicular to $\vec{b}$ has the magnitude $\frac{|\vec{a} \times \vec{b}|}{|\vec{b}|}$. The cross product is $(44, -24, -26)$ with the magnitude $\sqrt{1936 + 576 + 676} = 56.46$, and $|\vec{b}| = \sqrt{46} = 6.782$. So the magnitude is $\frac{56.46}{6.782} = 8.32$.