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 $DC$ and $AB$ is $\theta$, then $\cos \theta$ is equal to

• A. $\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$9$}\right.$
• B. $\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$9$}\right.$
• C. $\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$9$}\right.$

Answer 1

The correct option is B

$\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$9$}\right.$

Explanation for correct option:

Direction ratios of $DC$ are $\left(-1-1,0+2,4-5\right)=\left(-2,2,-1\right)$

Direction ratios of $BA$ are $\left(4-3,6-4,3-5\right)=\left(1,2,-2\right)$

As we know,

$\begin{array}{rcl}\cos \theta & =& \frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{{a_1}^2+{a_2}^2+{a_3}^2}\sqrt{{b_1}^2+{b_2}^2+{b_3}^2}}\\ & =& \frac{-2\times 1+2\times 2-1\left(-2\right)}{\sqrt{4+4+1}\sqrt{1+4+4}}\\ & =& \frac{\mathbf{4}}{\mathbf{9}}\end{array}$

Hence, Option ‘B’ is Correct.