Question

If $A=(1,2,3),B=(2,3,4),C=(1,-2,3)$ and the lines $AB,CD$ are perpendicular, then which of the following can be the coordinates of point $D$ ?

• A. $(2,4,-4)$
• B. $(2,2,-4)$
• C. $(4,2,-1)$
• D. $(-4,2,1)$

Answer 1

The correct option is A $(2,4,-4)$

Given $A=(1,2,3),B=(2,3,4),C=(1,-2,3)$
Let the coordinates of point $D$ be $(x,y,z)$
Direction ratios of $AB$ are $1,1,1$.
Direction ratios of $CD$ are $x-1,y+2,z-3$.
Now, since $AB$ and $CD$ are perpendicular
$\Rightarrow x-1+y+2+z-3=0$
$\Rightarrow x+y+z=2$
So, all the points satisfying this relation can be the coordinates of $D$.
Option A satisfies above relation.
Hence, coordinates of $D$ can be $(2,4,-4)$.