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GATE 2021 MN – Question 18

Engineering Mathematics · Vector Calculus · 1 mark · Multiple choice

For the coplanar vectors shown, let $\mathbf c=(\mathbf a+\mathbf b)\times(\mathbf a-\mathbf b)$. Its magnitude equals which quantity?

Diagram for GATE 2021 MN question 18
  1. Zero
  2. Half the area spanned by a and b
  3. The area spanned by a and b
  4. Twice the area spanned by a and b

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Show answer and explanation

Correct answer: (D) Twice the area spanned by a and b

Explanation

Expand by bilinearity. Self-cross-products vanish and $\mathbf b\times\mathbf a=-\mathbf a\times\mathbf b$, so $\mathbf c=-2(\mathbf a\times\mathbf b)$. Its magnitude is twice the parallelogram area spanned by the two vectors, D. The sign affects direction, not magnitude.

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