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Vector Algebra — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsVector Algebra1 Mark
MCQ
Assertion (A) : Three points with position vectors $\vec{a}, \vec{b}$ and $\vec{c}$ are collinear if $\vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}=\overrightarrow{0}$
Reason (R): If $\overrightarrow{A B} \cdot \overrightarrow{A C}=0$, then $\overrightarrow{A B} \perp \overrightarrow{A C}$.
  • A
    Both (A) and (R) are true and (R) is the correct explanation of (A).
  • ✓
    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: B.
Both (A) and (R) are true but (R) is not the correct explanation of (A).
(b) : If $A, B, C$ are collinear, then $\overrightarrow{A B}=k \overrightarrow{A C}$
$
\begin{array}{l}
\therefore \overrightarrow{A B} \times \overrightarrow{A C}=\overrightarrow{0} \Rightarrow(\vec{b}-\vec{a}) \times(\vec{c}-\vec{a})=\overrightarrow{0} \\
\Rightarrow \vec{b} \times \vec{c}+\vec{a} \times \vec{b}+\vec{c} \times \vec{a}=\overrightarrow{0} \text { i.e., } \vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}=\overrightarrow{0}
\end{array}
$
Hence, both assertion and reason are true but reason is not the correct explanation of assertion.

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