Question

The angle between the lines joining the points $(1,0,-3),(2,-1,2)$ and $(1,1,1),(3,2,0)$ is

• A. ${\cos }^{-1}\frac{2}{9}$
• B. ${90}^{\circ }$
• C. $0^{\circ }$
• D. ${\cos }^{-1}\frac{2\sqrt{2}}{9}$

Answer 1

The correct option is D ${\cos }^{-1}\frac{2\sqrt{2}}{9}$

Let $A(1,0,-3)B(2,-1,2)C(1,1,1)$ and $D(3,2,0)$
As we know that
cos$\theta$ $=$ dot product of direction cosine of AB and CD
direction cosine of AB$=$ ($\frac{1}{\sqrt{27}}$,$\frac{-1}{\sqrt{27}}$,$\frac{5}{\sqrt{27}}$)
Direction cosine of CD $=$ $\frac{2}{\sqrt{6}}$,$\frac{1}{\sqrt{6}}$,$\frac{-1}{\sqrt{6}}$)
cos$\theta$ $=$ $\frac{2-1-5}{\sqrt{27}\times \sqrt{6}}$
cos$\theta$ $=$ $\frac{-4}{\sqrt{27}\times \sqrt{6}}$
$\theta$ $=$ ${cos}^{-1}$$\frac{2\sqrt{2}}{9}$