Three vectors and are shown in the figure. Let S be any point on the vector . The distance between the points P and S is . The general relation among vectors and is [2017]

Question

Three vectors $\vec{P}, \vec{Q}$ and $\vec{R}$ are shown in the figure. Let S be any point on the vector $\vec{R}$ . The distance between the points P and S is $b | \vec{R} |$ . The general relation among vectors $\vec{P}, \vec{Q}$ and $\vec{S}$ is [2017]

Smart Achievers · Accepted Answer

(1)

$Here \vec{P} + b \vec{R} = \vec{S} \Rightarrow \vec{R} = \frac{\vec{S} - \vec{P}}{b}$

$Also \vec{R} = \vec{Q} - \vec{P}$

$∴$ $\frac{\vec{S} - \vec{P}}{b} = \vec{Q} - \vec{P} \Rightarrow \vec{S} - \vec{P} = b \vec{Q} - b \vec{P}$

$∴$ $\vec{S} = b \vec{Q} + (1 - b) \vec{P}$

Solution

Q.
Three vectors $\vec{P}, \vec{Q}$ and $\vec{R}$ are shown in the figure. Let S be any point on the vector $\vec{R}$ . The distance between the points P and S is $b | \vec{R} |$ . The general relation among vectors $\vec{P}, \vec{Q}$ and $\vec{S}$ is [2017]

1 $\vec{S} = (1 - b) \vec{P} + b \vec{Q}$

2 $\vec{S} = (b - 1) \vec{P} + b \vec{Q}$

3 $\vec{S} = (1 - b^{2}) \vec{P} + b \vec{Q}$

4 $\vec{S} = (1 - b) \vec{P} + b^{2} \vec{Q}$

Ans.
(1)

$Here \vec{P} + b \vec{R} = \vec{S} \Rightarrow \vec{R} = \frac{\vec{S} - \vec{P}}{b}$

$Also \vec{R} = \vec{Q} - \vec{P}$

$∴$ $\frac{\vec{S} - \vec{P}}{b} = \vec{Q} - \vec{P} \Rightarrow \vec{S} - \vec{P} = b \vec{Q} - b \vec{P}$

$∴$ $\vec{S} = b \vec{Q} + (1 - b) \vec{P}$

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Solution

Q. Three vectors P→,Q→ and R→ are shown in the figure. Let S be any point on the vector R→. The distance between the points P and S is b|R→|. The general relation among vectors P→,Q→ and S→ is [2017]

1 S→=(1-b)P→+bQ→

2 S→=(b-1)P→+bQ→

3 S→=(1-b2)P→+bQ→

4 S→=(1-b)P→+b2Q→

Ans. (1) Here P→+bR→=S→ ⇒ R→=S→-P→b Also R→=Q→-P→ ∴ S→-P→b=Q→-P→ ⇒ S→-P→=bQ→-bP→ ∴ S→=bQ→+(1-b)P→