GATE CSE 2017 Set 1 | Question: 30

Question

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 $v$?

• A. $2$
• B. $\frac{1}{2}$
• C. $1$
• D. $\frac{ -1}{2}$

Comments

I Know my question is stupid

What is the meaning of R2?

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R2 means the field of dimension 2 also we can say that if the vector is in R2 it will only have 2 dimensions that is [x,y] for R3 [x,y,z]

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There is a typing error in the question. it shall be “Let $u$ and $v$ be two vectors in $\mathbb{R}^2$” and not “Let $u$ and $v$ be two vectors in R2”

Answer 1

Statement :- If a and b be two vectors of equal length, then vector a+b will bisect the angle between a and b.

Proof : 

Here OBCA is a parallelogram. Consider triangle OAC and OBC, they both are congruent to each other (OA=OB, because vector a and b are of equal length ; AC=BC ; OC is common between them). So, angle AOC = angle BOC. Therefore we can conclude that OC bisects angle AOB. Hence Proved.

 

Since ||u|| = 2*||v|| and vector w bisects the angle between u & v. So, w must be u+2v. Therefore $\alpha$=2.