Question

If A and B are the points $(2,1,-2),(3,-4,5)$, then the angle that $OA$ makes with $OB$ is:

• A. ${\cos }^{-1}\left(\frac{4\sqrt{2}}{15}\right)$
• B. ${\cos }^{-1}\left(\frac{4}{15\sqrt{2}}\right)$
• C. ${\cos }^{-1}\left(\frac{18\sqrt{2}}{15}\right)$
• D. $\frac{\pi }{2}$

Answer 1

The correct option is A ${\cos }^{-1}\left(\frac{4\sqrt{2}}{15}\right)$

$0\left(0,0,0\right)$

$OA=-2\hat{i}-\hat{j}+2\hat{k}$, $\left(OA\right)=\sqrt{4+1+4}=3$

$OB=-3\hat{i}+4\hat{j}-5\hat{k}$, $\left(OB\right)=\sqrt{9+16+25}=\sqrt{5}0$

Now $\cos \theta =\left|\frac{\overrightarrow{OA}.\overrightarrow{OB}}{\left(\overrightarrow{OA}\right).\left(\overrightarrow{OB}\right)}\right|$

$=\left|\frac{6-4-10}{3\times \sqrt{50}}\right|$

$=\frac{8}{3\times 5\sqrt{2}}$

$\cos \theta =\frac{8}{15\sqrt{2}}$

$\cos \theta =\frac{8\times \sqrt{2}}{15\times 2}$

$\cos \theta =\frac{4\sqrt{2}}{15}$

$\theta ={\cos }^{-1}\left(\frac{4\sqrt{2}}{15}\right)$