Question

The distance of the point \( (1,-5,9) \), from the plane \( \vec{r}\cdot(\hat{i}-\hat{j}+\hat{k}) = 5 \) measured along the line \( \vec{r} = \hat{i} + \hat{j} + \hat{k} \) is

• A. \( 3\sqrt{5} \)
• B. \( 10\sqrt{3} \)
• C. \( 5\sqrt{3} \)
• D. \( 3\sqrt{10} \)

Hint:

For distance along a line, use projection of displacement vector on the direction vector.

Answer 1

The Correct Option is B

Concept: Distance along a line is found by projecting the vector onto the given direction.

Step 1: Plane normal vector: \[ \vec{n} = (1,-1,1) \] Line direction: \[ \vec{d} = (1,1,1) \]

Step 2: Use projection formula: \[ \text{Distance} = \frac{| \vec{n}\cdot \vec{OP} - 5 |}{|\vec{n}\cdot \hat{d}|} \] After simplification: \[ = 10\sqrt{3} \] Final Answer: \[ 10\sqrt{3} \]