The angle between a line with direction ratio | vedclass

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Question

(A) The direction ratios of the line joining the points $(3, 1, 4)$ and $(7, 2, 12)$ are given by $\langle 7-3, 2-1, 12-4 \rangle = \langle 4, 1, 8 \r

Vedclass · Accepted Answer

$\cos ^{-1}(2 / 3)$

Vedclass · Answer

(A) The direction ratios of the line joining the points $(3, 1, 4)$ and $(7, 2, 12)$ are given by $\langle 7-3, 2-1, 12-4 angle = \langle 4, 1, 8 angle$. Let these be $\langle a_1, a_2, a_3 angle = \langle 4, 1, 8 angle$. The direction ratios of the given line are $\langle b_1, b_2, b_3 angle = \langle 2, 2, 1 angle$. Let $	heta$ be the angle between the two lines. The formula for the cosine of the angle is $\cos 	heta = \frac{|a_1 b_1 + a_2 b_2 + a_3 b_3|}{\sqrt{a_1^2 + a_2^2 + a_3^2} \sqrt{b_1^2 + b_2^2 + b_3^2}}$. Substituting the values: $\cos 	heta = \frac{|(4)(2) + (1)(2) + (8)(1)|}{\sqrt{4^2 + 1^2 + 8^2} \sqrt{2^2 + 2^2 + 1^2}}$. $\cos 	heta = \frac{|8 + 2 + 8|}{\sqrt{16 + 1 + 64} \sqrt{4 + 4 + 1}} = \frac{18}{\sqrt{81} \sqrt{9}}$. $\cos 	heta = \frac{18}{9 	imes 3} = \frac{18}{27} = \frac{2}{3}$. Therefore,$	heta = \cos ^{-1}\left(\frac{2}{3}ight)$.

The angle between a line with direction ratios $2: 2: 1$ and a line joining the points $(3, 1, 4)$ and $(7, 2, 12)$ is

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