Question

$A=(2,3,-1),B=(3,5,-3),C=(1,2,3),D=(3,5,k)$ are four points. If the lines $AB$ and $CD$ are perpendicular, then $k=$

• A. $-7$
• B. $7$
• C. $2$
• D. $1$

Answer 1

The correct option is B $7$

The dot product of drs of $AB$ and $CD$ will be zero.
Therefore, $AB$ implies $(3-2,5-3,-3+1)$ $=(1,2,-2)$
$CD$ implies $(3-1,5-2,k-3)$ $=(2,3,k-3)$
Hence by dot product, we get
$\Rightarrow 2\left(1\right)+3\left(2\right)+(k-3)(-2)=0$
$\Rightarrow 2+6-2k+6=0$
$\Rightarrow 2k=14$
$\Rightarrow k=7$