Question

If \[ \vec r=\vec a+t\vec b \quad\text{and}\quad \vec r=\vec p+l\vec b \] are two lines, where \[ \vec a=\hat i+2\hat j,\qquad \vec b=\hat i+\hat j+\hat k,\qquad \vec p=2\hat i+\hat k. \] The position vector of a point \(A\) is \[ 5\hat i+\hat j-3\hat k. \] Let \(B,C\) be the feet of the perpendiculars drawn from \(A\) to the given two lines respectively, then \[ \overrightarrow{BA}+\overrightarrow{AC}= \]

• A. \(\hat i-2\hat j+3\hat k\)
• B. \(2\hat i-\hat j+3\hat k\)
• C. \(\hat i+\hat j+\hat k\)
• D. \(\hat i-2\hat j+\hat k\)

Hint:

The foot of the perpendicular from a point to a line is obtained by using the condition \[ \boxed{(\overrightarrow{PB})\cdot\vec d=0,} \] where \(\vec d\) is the direction vector of the line.

Answer 1

The Correct Option is D