Assertion (A) : If the points P=(a+b-c), Q=(2 a+b) and R=(b+t c) … - Vidyadip

MCQ

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
$\overrightarrow{P Q} \| \overrightarrow{P R} \text { or } \overrightarrow{P Q}=\lambda \overrightarrow{P R}, \lambda \in R$

✓
Both $(A)$ and$ (R)$ are true and $(R)$ is the correct explanation of $(A).$
B
Both $(A)$ and $(R)$ are true but $(R)$ is not the correct explanation of $(A).$
C
$(A)$ is true but $(R)$ is false.
D
$(A)$ is false but $(R)$ is true.

✓

Answer

Correct option: A.

Both $(A)$ and$ (R)$ are true and $(R)$ is the correct explanation of $(A).$

If $P, Q, R$ are collinear, then $\overrightarrow{P Q} \| \overrightarrow{P R}$ or $\overrightarrow{P Q}=\lambda \overrightarrow{P R}, \lambda \in R$
$\Rightarrow (2 \vec{a}+\vec{b})-(\vec{a}+\vec{b}-\vec{c})=\lambda[(\vec{b}+t \vec{c})-(\vec{a}+\vec{b}-\vec{c})]$
$\Rightarrow (\vec{a}+\vec{c})=\lambda[-\vec{a}+(t+1) \vec{c}]$
$\Rightarrow \vec{a}+\vec{c}=-\lambda \vec{a}+\lambda(t+1) \vec{c}$
On comparing, we get $-\lambda=1$
$\Rightarrow \lambda=-1$
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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