Question

If $A(3,4,5),B(4,6,3),C(-1,2,4)$ and $D(1,0,5)$ are such that the angle between the lines $\overline{DC}$ and $\overline{AB}$ is $\theta$ then $cos\theta =$

• A. $\frac{7}{9}$
• B. $\frac{2}{9}$
• C. $\frac{4}{9}$
• D. $\frac{5}{9}$

Answer 1

The correct option is C $\frac{4}{9}$

Given, $A(3,4,5),B(4,6,3),C(-1,2,4),D(1,0,5)$

$\overrightarrow{DC}=-2i+2j-k$...(i)
$\overrightarrow{AB}=i+2j-2k$ ...(ii)

Hence, $DC.AB\cos \theta =\overrightarrow{DC}.\overrightarrow{AB}$

$9\cos \theta =-2+4+2$
$\cos \theta =\frac{4}{9}$