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Three Dimensional Geometry — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsThree Dimensional Geometry1 Mark
MCQ
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 $\cos \theta=\frac{\vec{b}_1 \cdot \vec{b}_2}{\left|\vec{b}_1\right|\left|\vec{b}_2\right|}$.
  • A
    Both Assertion $(A)$ and Reason $(R)$ are true and Reason $(R)$ is the correct explanation of Assertion $(A).$
  • ✓
    Both Assertion $(A)$ and Reason $(R)$ are true and Reason $(R)$ is not the correct explanation of Assertion $(A).$
  • C
    Assertion $(A)$ is true but Reason $(R)$ is false.
  • D
    Assertion $(A)$ is false but Reason $(R)$ is true.
✓

Answer

Correct option: B.
Both Assertion $(A)$ and Reason $(R)$ are true and Reason $(R)$ is not the correct explanation of Assertion $(A).$
if lines are perpendicular, then $\theta=\frac{\pi}{2}$
$\therefore \cos \frac{\pi}{2}=\frac{\vec{b}_1 \cdot \vec{b}_2}{\left|\vec{b}_1\right| \vec{b}_2 \mid}$
$\Rightarrow \cos \frac{\pi}{2}=\frac{\vec{b}_1 \cdot \vec{b}_2}{\left|\vec{b}_1\right| \vec{b}_2 \mid}$
$\Rightarrow \vec{b}_1 \cdot \vec{b}_2=0$
$\therefore \quad$ Both $A$ and $R$ are true and $R$ is the correct explanation of $A$.

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