Question

If $A(0,4,1)$, $B(2,3,-1)$, $C(4,5,0)$, then find the angle between the lines $AB$ and $BC$.

• A. $0^{\circ }$
• B. $30^{\circ }$
• C. $45^{\circ }$
• D. $90^{\circ }$

Answer 1

Answer: (D)

$d.r{.}^{\prime }s$ of line $AB$ are $<2-0$, $3-4$, $-1-1>$
i.e., $<2$, $-1$, $-2>$.
$d.r{.}^{\prime }s$ of line $BC$ are $<4-2$, $5-3$, $0+1>$
i.e., $<2$, $2$, $1>$.
$\therefore cos\theta =\frac{\left(2\right)\left(2\right)+\left(-1\right)\left(2\right)+\left(-2\right)\left(1\right)}{\sqrt{4+1+4}\sqrt{4+4+1}}=0$
$\Rightarrow \theta =90^{\circ }$