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GATE 2022 MN – Question 14

Engineering Mathematics · Vector Calculus · 1 mark · Multiple choice

For the vectors x and y shown, which expression is the vector projection of x onto nonzero y?

Diagram for GATE 2022 MN question 14
  1. $\frac{\mathbf x^T\mathbf y}{\mathbf y^T\mathbf y}\mathbf y$
  2. $\mathbf x\times\mathbf y$
  3. $\frac{\mathbf x\times\mathbf y}{\mathbf y^T\mathbf y}$
  4. $\frac{\mathbf x^T\mathbf y}{\mathbf x^T\mathbf x}\mathbf x$

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Correct answer: (A) $\frac{\mathbf x^T\mathbf y}{\mathbf y^T\mathbf y}\mathbf y$

Explanation

The projection must be parallel to y, so write it as cy. Orthogonality of the residual requires (x−cy)ᵀy=0, giving c=(xᵀy)/(yᵀy). This yields A. A cross product is perpendicular to the plane, and D instead projects onto x.

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