If 1,−1,1 and 2,−3,5 are direction ratios of two lines, then the angle between them is

# If $\left(1,-1,1\right)$ and $\left(2,-3,5\right)$ are direction ratios of two lines, then the angle between them is

1. A

${\mathrm{cos}}^{-1}\left(\frac{5\sqrt{2}}{\sqrt{57}}\right)$

2. B

${\mathrm{sin}}^{-1}\left(\frac{5\sqrt{2}}{\sqrt{57}}\right)$

3. C

${\mathrm{cos}}^{-1}\left(\frac{50}{\sqrt{114}}\right)$

4. D

${\mathrm{tan}}^{-1}\left(\frac{5\sqrt{2}}{\sqrt{57}}\right)$

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### Solution:

If $\theta$ is the acute angle between two lines having direction ratios $〈{a}_{1},{b}_{1},{c}_{1}〉$and $〈{a}_{2},{b}_{2},{c}_{2}〉$

then $\mathrm{cos}\theta =\left|\frac{{a}_{1}{a}_{2}+{b}_{1}{b}_{2}+{c}_{1}{c}_{2}}{\sqrt{{{a}_{1}}^{2}+{{b}_{1}}^{2}+{{c}_{1}}^{2}}\cdot \sqrt{{{a}_{2}}^{2}+{{b}_{2}}^{2}+{{c}_{2}}^{2}}}\right|$

Hence,

$\begin{array}{c}\mathrm{cos}\theta =\frac{2+3+5}{\sqrt{1+1+1}\cdot \sqrt{4+9+25}}\\ =\frac{10}{\sqrt{3}\cdot \sqrt{38}}\\ =\frac{5\sqrt{2}}{\sqrt{57}}\end{array}$

Therefore,$\theta =\overline{){\mathrm{cos}}^{-1}\left(\frac{5\sqrt{2}}{\sqrt{57}}\right)}$

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