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If $\theta$ is the angle between the lines $\mathrm{AB}$ and $\mathrm{AC}$ where $A, B$ and $C$ are the three points with coordinates $(1,2,-1),(2,0,3),(3,-1,2)$ respectively, then $\sqrt{462} \cos \theta$ is equal to
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The correct answer is:
20
Given : A, B \& C are three points with coordinates $(1,2,-1),(2,0,3) \&(3,-1,2)$ respectively.
Now, direction ratio's of $\mathrm{AB}=2-1,0-2,3+1=1,-2,4 \&$ direction ratio's of $\mathrm{AC}=3-1,-1-2,2+1=2,-3,3$ we know that
$\cos \theta=\frac{\mathrm{a}_{1} \mathrm{a}_{2}+\mathrm{b}_{1} \mathrm{~b}_{2}+\mathrm{c}_{1} \mathrm{c}_{2}}{\sqrt{\mathrm{a}_{1}{ }^{2}+\mathrm{b}_{1}{ }^{2}+\mathrm{c}_{1}{ }^{2}} \cdot \sqrt{\mathrm{a}_{2}{ }^{2}+\mathrm{b}_{2}{ }^{2}+\mathrm{c}_{2}{ }^{2}}}$
$\Rightarrow \cos \theta=\frac{(1)(2)+(-2)(-3)+(4)(3)}{\sqrt{1+4+16} \cdot \sqrt{4+9+9}}$
$=\frac{2+6+12}{\sqrt{21} \cdot \sqrt{22}}=\frac{20}{\sqrt{462}}$
$\Rightarrow \sqrt{462} \cos \theta=20$
Now, direction ratio's of $\mathrm{AB}=2-1,0-2,3+1=1,-2,4 \&$ direction ratio's of $\mathrm{AC}=3-1,-1-2,2+1=2,-3,3$ we know that
$\cos \theta=\frac{\mathrm{a}_{1} \mathrm{a}_{2}+\mathrm{b}_{1} \mathrm{~b}_{2}+\mathrm{c}_{1} \mathrm{c}_{2}}{\sqrt{\mathrm{a}_{1}{ }^{2}+\mathrm{b}_{1}{ }^{2}+\mathrm{c}_{1}{ }^{2}} \cdot \sqrt{\mathrm{a}_{2}{ }^{2}+\mathrm{b}_{2}{ }^{2}+\mathrm{c}_{2}{ }^{2}}}$
$\Rightarrow \cos \theta=\frac{(1)(2)+(-2)(-3)+(4)(3)}{\sqrt{1+4+16} \cdot \sqrt{4+9+9}}$
$=\frac{2+6+12}{\sqrt{21} \cdot \sqrt{22}}=\frac{20}{\sqrt{462}}$
$\Rightarrow \sqrt{462} \cos \theta=20$
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