Single Choice

The acute angle between two lines whose direction ratios are $$2,3,6$$ and $$1,2,2$$ is

A$$\displaystyle \cos ^{ -1 }{ \left( \frac { 20 }{ 21 } \right) } $$
Correct Answer
B$$\displaystyle \cos ^{ -1 }{ \left( \frac { 18 }{ 21 } \right) } $$
C$$\displaystyle \cos ^{ -1 }{ \left( \frac { 8 }{ 21 } \right) } $$
DNone of these

Solution

We have $${ a }_{ 1 }=2,{ b }_{ 1 }=3,{ c }_{ 1 }=6,{ a }_{ 2 }=1,{ b }_{ 2 }=2,{ c }_{ 2 }=2$$
If $$\theta$$ is the angle between two lines whose direction ratios are given, then
$$\cos { \theta } =\displaystyle\frac { { a }_{ 1 }{ a }_{ 2 }+{ b }_{ 1 }{ b }_{ 2 }+{ c }_{ 1 }{ c }_{ 2 } }{ \sqrt { { \left( { a }_{ 1 } \right) }^{ 2 }+{ \left( { b }_{ 1 } \right) }^{ 2 }+{ \left( { c }_{ 1 } \right) }^{ 2 } } \sqrt { { \left( { a }_{ 2 } \right) }^{ 2 }+{ \left( { b }_{ 2 } \right) }^{ 2 }+{ \left( { c }_{ 2 } \right) }^{ 2 } } } $$
$$\Rightarrow \cos { \theta } =\displaystyle \frac { 2\times 1+3\times 2+6\times 2 }{ \sqrt { { \left( 2 \right) }^{ 2 }+{ \left( 3 \right) }^{ 2 }+{ \left( 6 \right) }^{ 2 } } \sqrt { { \left( 1 \right) }^{ 2 }+{ \left( 2 \right) }^{ 2 }+{ \left( 2 \right) }^{ 2 } } } =\frac { 20 }{ 21 }$$
$$ \Rightarrow \theta =\cos ^{ -1 }{ \dfrac { 20 }{ 21 } } $$

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