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GATE 2025 CH – Question 12

Engineering Mathematics · Calculus: Gradient, divergence, curl, vector identities and integral theorems · 1 mark · Multiple choice

Consider a Cartesian coordinate system defined over a 3-dimensional vector space with orthogonal unit basis vectors $\hat{i}$, $\hat{j}$ and $\hat{k}$. Let vector $\mathbf{a} = \sqrt{2}\hat{i} + \frac{1}{\sqrt{2}}\hat{j} + \hat{k}$, and vector $\mathbf{b} = \frac{1}{\sqrt{2}}\hat{i} + \sqrt{2}\hat{j} - \hat{k}$. The inner product of these vectors ($\mathbf{a} \cdot \mathbf{b}$) is

  1. 0
  2. 1
  3. −1
  4. 2

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Correct answer: (B) 1

Explanation

$\mathbf{a} \cdot \mathbf{b} = \sqrt{2} \cdot \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} \cdot \sqrt{2} + (1)(-1) = 1 + 1 - 1 = 1$.