in Linear Algebra edited by
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30 votes
30 votes

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 $v$?

  1. $2$
  2. $\frac{1}{2}$
  3. $1$
  4. $\frac{ -1}{2}$
in Linear Algebra edited by
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4 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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4

 

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”

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5 Answers

25 votes
25 votes
Best answer
Angle between $u$ and $w =$ Angle between $w$ and $v$

$\frac{\vec{u}. \vec{w}}{\|u\| \|w\| } = \frac{\vec{w}. \vec{v}}{\|w\| \|v\| }$

$\vec{u}. \vec{w} = 2 \vec{w}. \vec{v}$

$(\alpha -2)\vec{u}. \vec{v} = 2(\alpha -2) \|v\|^2$

$\text{LHS}$ and $\text{RHS}$ would be equal for $\alpha=2$. Hence, correct answer is $(A)$.
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4 Comments

no need it is outta syllaba
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@ijnuhb, In 2021 also there was one question from normal vector to plane, So should we be prepared for this topic too?

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@ijnuhb r u sure it’s not in syllabus?

if it’s not in syllabus then @Lakshman Patel RJIT sir plz tag this qn as out of syllabus cz it’s creating confusion!!

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22 votes
22 votes

We know that  " The resultant of two equal vectors bisects the angle between them ".

In the given question, $$\overrightarrow{w}  = \overrightarrow{u}  + \alpha \overrightarrow{v} $$

Here, $\overrightarrow{w}$ is the resultant of two vectors $\overrightarrow{u}$ and $\alpha$$\overrightarrow{v}$. 

Thus, for $\overrightarrow{w}$ to bisect the angle between $\overrightarrow{u}$ and $\overrightarrow{v}$, magnitude of the two vectors must be equal.

                                      $\left \| u \right \| =\left \| \alpha v \right \|$               $\Rightarrow$           $2\left \| v \right \| =\alpha \left \| v \right \|$

$\therefore$         $\alpha = 2$

8 votes
8 votes

or can be done simply using vector diagrams.

3 votes
3 votes

The geometric shape can also be parallelogram

1 comment

in ur diagram i think ... the bisector would be   2u+v    instead of u+2v... ...m i correct?

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Answer:

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