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Vector Algebra — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsVector Algebra1 Mark
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).
(a) : If $P, Q, R$ are collinear, then $\overrightarrow{P Q} \| \overrightarrow{P R}$ or $\overrightarrow{P Q}=\lambda \overrightarrow{P R}, \lambda \in R$
$
\begin{array}{l}
\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 \quad \vec{a}+\vec{c}=-\lambda \vec{a}+\lambda(t+1) \vec{c}
\end{array}
$
On comparing, we get
$
\begin{array}{l}
-\lambda=1 \Rightarrow \lambda=-1 \\
\text { and } \lambda(t+1)=1 \Rightarrow-(t+1)=1 \\
\Rightarrow-t-1=1 \Rightarrow t=-2
\end{array}
$
Hence, both assertion are true and reason is the correct explanation of assertion.

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