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GATE 2017 CS – Question 40

Engineering Mathematics · Linear Algebra · 2 marks · Multiple choice

Let $u$ and $v$ be two vectors in $\mathbb{R}^2$ whose Euclidean norms satisfy $\|u\| = 2\|v\|$. What is the value of $\alpha$ such that $w = u + \alpha v$ bisects the angle between $u$ and $v$?

  1. 2
  2. $\frac{1}{2}$
  3. 1
  4. $-\frac{1}{2}$

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Correct answer: (A) 2

Explanation

The bisector of two vectors points along the sum of their unit vectors, $\frac{u}{\|u\|} + \frac{v}{\|v\|} = \frac{u}{2\|v\|} + \frac{v}{\|v\|}$. Multiplying by $2\|v\|$ gives $u + 2v$, so $\alpha = 2$.