Question

If the direction ratios of two mutually perpendicular lines are \(2,\,3,\,5\) and \(x,\,y,\,4\), then \(2x+3y=\) ?

• A. \(20\)
• B. \(-20\)
• C. \(30\)
• D. \(-30\)

Hint:

Perpendicular \(\Rightarrow\) dot product \(=0\): multiply component-wise and add.

Answer 1

The Correct Option is B

Idea. Two lines are perpendicular in 3-D exactly when the dot product of any direction-ratio triples is \(0\). Dot product adds the products of the matching components.
Step 1. Compute the dot product and set it to zero. \[ (2,3,5)\cdot(x,y,4)=2x+3y+5\cdot 4=0. \] Step 2. Simplify. \[ 2x+3y+20=0\Rightarrow 2x+3y=-20. \] That is the required value.