Allied & ICAR Exam Bank • ALLIED_2671
MCQ • 1 Mark
Section 1: Engineering Mathematics
Vector Calculus
Subtopic: Scalar and vector point functions
Question Statement:
A unit normal vector to the level surface $x^2 + y^2 + z^2 = 3$ at the point $(1, 1, 1)$ is:
Options:
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(A) $\frac{1}{\sqrt{3}}(\hat{i} + \hat{j} + \hat{k})$
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(B) $\frac{1}{3}(\hat{i} + \hat{j} + \hat{k})$
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(C) $\hat{i} + \hat{j} + \hat{k}$
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(D) $\frac{1}{\sqrt{2}}(\hat{i} + \hat{j})$
Step-by-Step Solution & Explanation
$\nabla f = 2x\hat{i} + 2y\hat{j} + 2z\hat{k} = 2(\hat{i} + \hat{j} + \hat{k})$ at $(1, 1, 1)$. Unit normal is $\frac{\nabla f}{|\nabla f|} = \frac{1}{\sqrt{3}}(\hat{i} + \hat{j} + \hat{k})$.
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