GATE CS 2017 Set 1 Q.30 (CS Q.30): Linear Algebra

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

-1/2

Official answer and solution

Official answer: A
From: Official GATE 2017 answer key

Solution

Express the angle bisector of two vectors using unit vectors and solve for the scalar coefficient $\alpha$ .

Let $u$ and $v$ be non-zero vectors in $\mathbb{R}^2$ with angle $\theta$ between them. A vector $w$ bisects the angle between $u$ and $v$ if it makes equal angles with $u$ and $v$ :

\frac{u \cdot w}{\|u\| \|w\|} = \frac{v \cdot w}{\|v\| \|w\|}

Geometrically, the sum of unit vectors $\hat{u} + \hat{v} = \frac{u}{\|u\|} + \frac{v}{\|v\|}$ points along the angle bisector. Any positive scalar multiple of this sum is also an angle bisector vector:

w = k \left(\frac{u}{\|u\|} + \frac{v}{\|v\|}\right)

Choosing $k = \|u\|$ to match the coefficient 1 for $u$ :

w = u + \frac{\|u\|}{\|v\|} v

Comparing with $w = u + \alpha v$ , we have:

\alpha = \frac{\|u\|}{\|v\|}

We are given that $\|u\| = 2 \|v\|$ , which implies:

\alpha = \frac{2 \|v\|}{\|v\|} = 2

Thus, $\alpha = 2$ .

Final answer

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2017 Computer Science and Information Technology (CS) Session 1
Question: 30 of 65, printed as CS Q.30
Section: Computer Science and Information Technology

Report an error in this question

GATE questions are the work of the organising institutes. This site is independent and is not affiliated with them.

Question

Solution

Source