Question

If $A=(1,2,-1),B=(2,0,3),C=(3,-1,2)$ then the angle between $\overline{AB}$ and $\overline{AC}$ is

• A. $0^o$
• B. ${90}^o$
• C. ${\cos }^{-1}\left(\frac{20}{\sqrt{21}\sqrt{22}}\right)$
• D. ${\cos }^{-1}\left(\frac{15}{\sqrt{21}\sqrt{11}}\right)$

Answer 1

The correct option is C ${\cos }^{-1}\left(\frac{20}{\sqrt{21}\sqrt{22}}\right)$

Given $A(1,2,-1),B=(2,0,3),C=(3,-1,2)$

Drs of $AB=(1,-2,4)$

Drs of $AC=(2,-3,3)$

$\cos \theta =\frac{2+6+12}{\sqrt{1+4+16}\sqrt{4+9+9}}$

$\Rightarrow$ $\cos \theta =\frac{20}{\sqrt{21}\sqrt{22}}$

$\Rightarrow$ $\theta ={\cos }^{-1}\left(\frac{20}{\sqrt{21}\sqrt{22}}\right)$