Question

Direction ratios of lines \(l_1\) and \(l_2\) respectively are \(\langle 1, 0, 0\rangle\) and \(\langle 0, -1, 0\rangle\). The direction ratios of the line perpendicular to both \(l_1\) and \(l_2\) are :

• A. \(\langle 1, 1, 0\rangle\)
• B. \(\langle 0, 0, -1\rangle\)
• C. \(\langle 1, 1, 1\rangle\)
• D. \(\langle 1, 0, -1\rangle\)

Hint:

Take the cross product of the two direction vectors. Only the k component survives.

Answer 1

The Correct Option is B

Step 1: Set up the conditions:
Let the required direction ratios be $\langle a, b, c\rangle$.
It is perpendicular to a line when the dot product of the two direction ratios is zero.

Step 2: Use the line $l_1$:
The dot product with $\langle 1,0,0\rangle$ is $a\cdot 1+b\cdot 0+c\cdot 0=a$.
Setting this to zero gives $a=0$.

Step 3: Use the line $l_2$:
The dot product with $\langle 0,-1,0\rangle$ is $-b$.
Setting this to zero gives $b=0$.
So the direction is $\langle 0,0,c\rangle$ with $c\neq 0$.

Step 4: Compare with the options:
Option (A) has $b=1$, so it fails.
Option (C) has $a=1$ and $b=1$, so it fails.
Option (D) has $a=1$, so it fails.
Option (B) is $\langle 0,0,-1\rangle$, which has $a=b=0$ and passes.
This also matches the cross product $\hat{i}\times(-\hat{j})=-\hat{k}$.

Final Answer:
Option (B) is the only option that works. \[ \boxed{\langle 0, 0, -1\rangle} \]