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Vector Algebra — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsVector Algebra1 Mark
MCQ
Assertion (A) : If $\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0},|\vec{a}|=3$, $|\vec{b}|=4,|\vec{c}|=5$, then $\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}$ is equal to -25 .
Reason (R) : If $\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}$, then the angle $\theta$ between $\vec{b}$ and $\vec{c}$ is given by
$
\cos \theta=\frac{|\vec{a}|^2-|\vec{b}|^2-|\vec{c}|^2}{2|\vec{b}||\vec{c}|^2} .
$
  • 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) : We have, $|\vec{a}|=3,|\vec{b}|=4,|\vec{c}|=5$ and
$
\begin{array}{l}
\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0} \Rightarrow(\vec{a}+\vec{b}+\vec{c})^2=0 \\
\Rightarrow|\vec{a}|^2+|\vec{b}|^2+|\vec{c}|^2+2(\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a})=0 \\
\Rightarrow(3)^2+(4)^2+(5)^2+2(\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a})=0
\end{array}
$
$
\begin{array}{l}
\Rightarrow \vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}=-\frac{1}{2}[9+16+25]=-\frac{1}{2}(50)=-25 \\
\text { Now, } \vec{a}+\vec{b}+\vec{c}=\overrightarrow{0} \Rightarrow \vec{b}+\vec{c}=-\vec{a} \\
\Rightarrow(\vec{b}+\vec{c})^2=(-\vec{a})^2 \Rightarrow|\vec{b}|^2+|\vec{c}|^2+2 \vec{b} \cdot \vec{c}=|\vec{a}|^2 \\
\Rightarrow|\vec{b}|^2+|\vec{c}|^2+2|\vec{b}||\vec{c}| \cos \theta=|\vec{a}|^2 \\
\Rightarrow \cos \theta=\frac{|\vec{a}|^2-|\vec{b}|^2-|\vec{c}|^2}{2|\vec{b}||\vec{c}|}
\end{array}
$
Hence, both assertion and reason are true but reason is not the correct explanation of assertion.

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