The GATE Grind

GATE 2023 EC – Question 11

Engineering Mathematics · Linear Algebra · 1 mark · Multiple choice

Let $\boldsymbol v_1=\begin{bmatrix}1\\2\\0\end{bmatrix}$ and $\boldsymbol v_2=\begin{bmatrix}2\\1\\3\end{bmatrix}$ be two vectors. The value of the coefficient $\alpha$ in the expression $\boldsymbol v_1=\alpha\boldsymbol v_2+\boldsymbol e$, which minimizes the length of the error vector $\boldsymbol e$, is

  1. $\dfrac72$
  2. $\dfrac{-2}{7}$
  3. $\dfrac27$
  4. $\dfrac{-7}{2}$

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Correct answer: (C) $\dfrac27$

Explanation

The best $\alpha$ makes $\boldsymbol e\perp\boldsymbol v_2$: $\alpha=\dfrac{\boldsymbol v_1\cdot\boldsymbol v_2}{\boldsymbol v_2\cdot\boldsymbol v_2}=\dfrac{2+2+0}{4+1+9}=\dfrac4{14}=\dfrac27$.