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NCERT Class XII Mathematics Chapter - - Solutions

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Question : 18 of 78
Marks: +1, -0
Find the angle between the following pair of lines:
x2=y2=z1\frac{x}{2}=\frac{y}{2}=\frac{z}{1} and x54=y21=z38\frac{x-5}{4}=\frac{y-2}{1}=\frac{z-3}{8}
Solution:  
Direction ratios of the given lines are respectively < 2, 2, 1 > and < 4, 1, 8 >
Let θ be the angle between the given lines, then
cosθ=a1a2+b1b2+c1c2a12+b12+c12a22+b22+c22,\cos \theta = \left| \frac{a_1a_2+b_1b_2+c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}} \right|,
where a1,b1,c1a_1, b_1, c_1 and a2,b2,c2a_2, b_2, c_2 are direction ratios
cosθ\cos \theta =2×4+2×1+1×822+22+1242+12+82= \frac{\left| 2\times 4+2\times 1+1\times 8 \right|}{\sqrt{2^2+2^2+1^2} \sqrt{4^2+1^2+8^2}}
=18981=23= \frac{18}{\sqrt{9} \sqrt{81}} = \frac{2}{3}
θ=cos123\Rightarrow \theta = \cos^{-1} \frac{2}{3}
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