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Vector Algebra — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSVector Algebra1 Mark
Question
Assertion $(A) :$ If the points $\vec{P}=(\vec{a}+\vec{b}-\vec{c})$, $\vec{Q}=(2 \vec{a}+\vec{b})$ and $\vec{R}=(\vec{b}+t \vec{c})$ are collinear, where $\vec{a}, \vec{b}, \vec{c}$ are three non-coplanar vectors, then the value of $t$ is $-2 .$
Reason $(R) :$ If $P, Q, R$ are collinear, then
$\overrightarrow{P Q} \| \overrightarrow{P R} \text { or } \overrightarrow{P Q}=\lambda \overrightarrow{P R}, \lambda \in R$
✓

Answer

$(a) :$ If $P, Q, R$ are collinear, then $\overrightarrow{P Q} \| \overrightarrow{P R}$ or $\overrightarrow{P Q}=\lambda \overrightarrow{P R}, \lambda \in R$
$\Rightarrow \quad(2 \vec{a}+\vec{b})-(\vec{a}+\vec{b}-\vec{c})=\lambda[(\vec{b}+t \vec{c})-(\vec{a}+\vec{b}-\vec{c})]$
$\Rightarrow \quad(\vec{a}+\vec{c})=\lambda[-\vec{a}+(t+1) \vec{c}]$
$\Rightarrow \vec{a}+\vec{c}=-\lambda \vec{a}+\lambda(t+1) \vec{c}$
On comparing, we get
$-\lambda=1$
$\Rightarrow \lambda=-1$
$\text { and } \lambda(t+1)=1 \Rightarrow-(t+1)=1$
$\Rightarrow-t-1=1$
$\Rightarrow t=-2$
Hence, both assertion are true and reason is the correct explanation of assertion.

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