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Three Dimensional Geometry — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSThree Dimensional Geometry1 Mark
Question
Assertion (A) : The lines $\vec{r}=\vec{a}_1+\lambda \vec{b}_1$ and $\vec{r}=\vec{a}_2+\mu \vec{b}_2$ are perpendicular, when $\vec{b}_1 \cdot \vec{b}_2=0$.
Reason (R): The angle $\theta$ between the lines $\vec{r}=\vec{a}_1+\lambda \vec{b}_1$ and $\vec{r}=\vec{a}_2+\mu \vec{b}_2$ is given by $\cos \theta=\frac{\vec{b}_1 \cdot \vec{b}_2}{\left|\vec{b}_1\right|\left|\vec{b}_2\right|}$.
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Answer

(a) : If lines are perpendicular, then $\theta=\frac{\pi}{2}$
$
\therefore \quad \cos \frac{\pi}{2}=\frac{\vec{b}_1 \cdot \vec{b}_2}{\left|\vec{b}_1\right| \vec{b}_2 \mid} \Rightarrow \cos \frac{\pi}{2}=\frac{\vec{b}_1 \cdot \vec{b}_2}{\left|\vec{b}_1\right| \vec{b}_2 \mid}
$
$
\Rightarrow \quad \vec{b}_1 \cdot \vec{b}_2=0
$
$\therefore \quad$ Both $A$ and $R$ are true and $R$ is the correct explanation of $A$.

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