Question

Assertion (A): For the lines $\mathbf{r} = \mathbf{a} + t \mathbf{b}$ and $\mathbf{r} = \mathbf{p} + s \mathbf{q}$, if $(\mathbf{a} - \mathbf{p}) \cdot (\mathbf{b} \times \mathbf{q}) \neq 0$, then the two lines are coplanar. Reason (R): $|(\mathbf{a} - \mathbf{p}) \cdot (\mathbf{b} \times \mathbf{q})|$ is $|\mathbf{b} \times \mathbf{q}|$ times the shortest distance between the lines $\mathbf{r} = \mathbf{a} + t \mathbf{b}$ and $\mathbf{r} = \mathbf{p} + s \mathbf{q}$.

• (A) is true, (R) is true, and (R) is the correct explanation to (A)
• (A) is true, (R) is true, and (R) is not the correct explanation to (A)
• (A) is true, (R) is false
• (A) is false, (R) is true

Hint:

Lines are coplanar if $(\mathbf{a} - \mathbf{p}) \cdot (\mathbf{b} \times \mathbf{q}) = 0$. The shortest distance formula uses the scalar triple product divided by $|\mathbf{b} \times \mathbf{q}|$.

Answer 1

The Correct Option is D

For two lines $\mathbf{r} = \mathbf{a} + t \mathbf{b}$ and $\mathbf{r} = \mathbf{p} + s \mathbf{q}$ to be coplanar, the scalar triple product must be zero: \[ (\mathbf{a} - \mathbf{p}) \cdot (\mathbf{b} \times \mathbf{q}) = 0 \] If $(\mathbf{a} - \mathbf{p}) \cdot (\mathbf{b} \times \mathbf{q}) \neq 0$, the lines are not coplanar (skew). Thus, Assertion (A) is false. The shortest distance between two skew lines is: \[ d = \frac{|(\mathbf{a} - \mathbf{p}) \cdot (\mathbf{b} \times \mathbf{q})|}{|\mathbf{b} \times \mathbf{q}|} \] Thus, $|(\mathbf{a} - \mathbf{p}) \cdot (\mathbf{b} \times \mathbf{q})| = |\mathbf{b} \times \mathbf{q}| \cdot d$, so Reason (R) is true. Since (A) is false, (R) cannot explain (A). Option (4) is correct. Options (1), (2), and (3) are incorrect due to the falsity of (A).