Assertion (A) : The lines r=a_1+ b_1 and r=a_2+ b_2 are perpendic… - Vidyadip

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 $\cos \theta=\frac{\vec{b}_1 \cdot \vec{b}_2}{\left|\vec{b}_1\right|\left|\vec{b}_2\right|}$.

✓
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 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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