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# Solution for Write the Angle Between the Lines Whose Direction Ratios Are Proportional to 1, −2, 1 and 4, 3, 2. - CBSE (Commerce) Class 12 - Mathematics

ConceptDirection Cosines and Direction Ratios of a Line

#### Question

Write the angle between the lines whose direction ratios are proportional to 1, −2, 1 and 4, 3, 2.

#### Solution

$\text{ The direction ratios of the first line are 1, - 2, 1 and the direction ratios of the second line are 4, 3, 2 } .$

$\text{ Let } \theta \text{ be the angle between these two lines } .$

$\text{ Now },$

$\cos \theta = \left| \frac{1\left( 4 \right) + \left( - 2 \right)\left( 3 \right) + 1\left( 2 \right)}{\sqrt{\left( 1 \right)^2 + \left( - 2 \right)^2 + \left( 1 \right)^2} \sqrt{\left( 4 \right)^2 + \left( 3 \right)^2 + \left( 2 \right)^2}} \right|$

$= \left| \frac{4 - 6 + 2}{\sqrt{1 + 4 + 1}\sqrt{16 + 9 + 4}} \right|$

$= \frac{0}{\sqrt{6}\sqrt{29}}$

$= 0$

$\Rightarrow \theta = \frac{\pi}{2}$

$\text { Hence, the required angle is } \frac{\pi}{2} .$

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Solution Write the Angle Between the Lines Whose Direction Ratios Are Proportional to 1, −2, 1 and 4, 3, 2. Concept: Direction Cosines and Direction Ratios of a Line.
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