Three vectors P → , Q → and R → are shown in the figure. Let, S be any point on the vector R → . The…

1. $\overrightarrow{S}=\left(1-b\right) \overrightarrow{P}+b^2\overrightarrow{Q}$
2. $\overrightarrow{S}=\left(B-1\right)\overrightarrow{P}+b\overrightarrow{Q}$
3. $\vec{S} = |1 - b|\vec{P} + b\vec{Q}$
4. $\overrightarrow{S}=\left(1-b^2\right)\overrightarrow{P}+b\overrightarrow{Q}$

Solution

Let, vector from point $P$ to point $S$ be $\overrightarrow{C}$, 

$\Rightarrow \overrightarrow{C}=b\left|\overrightarrow{R}\right| \hat{R}=b\left|\overrightarrow{R}\right| \left(\frac{\overrightarrow{R}}{\left|\overrightarrow{R}\right|}\right)=b\overrightarrow{R}=b\left(\overrightarrow{Q}- \overrightarrow{P}\right)$

From triangle rule of vector addition, 

$\overrightarrow{P}+ \overrightarrow{C}= \overrightarrow{S}$

$\overrightarrow{P}+b\left(\overrightarrow{Q}- \overrightarrow{P}\right)= \overrightarrow{S}$

$\Rightarrow \overrightarrow{S}=\left(1-b\right)\overrightarrow{P}+b\overrightarrow{Q}$

Asked in: JEE Advanced 2017 (Paper 2)

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