Official PYQ · 2017Set 1 · Q30Linear Algebra: Matrices, Determinants, System of Linear EquationsMCQ1 mark

GATE CS 2017 Set 1 Q30 — Linear Algebra: Matrices, Determinants, System of Linear Equations

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

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Answer: A

An elegant way to solve this is using unit vectors.

Let $\hat{u}$ and $\hat{v}$ be the unit vectors in the directions of $u$ and $v$ respectively:

\hat{u} = \frac{u}{\|u\|} \quad \text{and} \quad \hat{v} = \frac{v}{\|v\|}

The vector that bisects the angle between $u$ and $v$ must point in the direction of the sum of their unit vectors, i.e., it must be a positive scalar multiple of $(\hat{u} + \hat{v})$ :

w = c(\hat{u} + \hat{v}) \quad \text{for some } c > 0

We are given that the Euclidean norms satisfy $\|u\| = 2\|v\|$ . Therefore, we can express $u$ and $v$ as:

u = \|u\|\hat{u} = 2\|v\|\hat{u}

v = \|v\|\hat{v}

Substitute these into the given expression for $w$ :

w = u + \alpha v

w = 2\|v\|\hat{u} + \alpha \|v\|\hat{v}

w = \|v\| (2\hat{u} + \alpha\hat{v})

For $w$ to bisect the angle between $u$ and $v$ , it must be a scalar multiple of $(\hat{u} + \hat{v})$ . This requires the coefficients of $\hat{u}$ and $\hat{v}$ to be equal:

2 = \alpha

Thus, $\alpha = 2$ .

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Published 10 Oct 2026

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