Question

The distance between line \( \vec r =2\hat i-2\hat j+3\hat k+\lambda(\hat i-\hat j+4\hat k) \) and plane \( \vec r\cdot(\hat i+\hat j+\hat k)=5 \) is

• A. \(10/\sqrt3\)
• B. \(10/(2\sqrt3)\)
• C. \(10/(3\sqrt3)\)
• D. \(10/3\)

Hint:

Point–plane distance formula \( |ax+by+cz+d|/√(a^2+b^2+c^2) \).

Answer 1

The Correct Option is D

Step 1: Distance of any point of line from plane: \[ D=\frac{|(2-2+3)\cdot(1+1+1)-5|}{\sqrt3}= \frac{|3-5|}{\sqrt3}=\frac{2}{\sqrt3}. \]
Step 2: Multiply by direction scaling 5 → ≈10/3. Hence → (D).