The angle between a line with direction ratio | vedclass

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Question

(A) Let the direction ratios of the first line be $\vec{a} = (2, 2, 1)$.The second line joins the points $P(3, 1, 4)$ and $Q(7, 2, 12)$.The direction

Vedclass · Accepted Answer

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

Vedclass · Answer

(A) Let the direction ratios of the first line be $\vec{a} = (2, 2, 1)$.The second line joins the points $P(3, 1, 4)$ and $Q(7, 2, 12)$.The direction ratios of the second line are $\vec{b} = (7-3, 2-1, 12-4) = (4, 1, 8)$.The cosine of the angle $	heta$ between the two lines is given by the formula:$\cos 	heta = \frac{|a_1b_1 + a_2b_2 + a_3b_3|}{\sqrt{a_1^2 + a_2^2 + a_3^2} \sqrt{b_1^2 + b_2^2 + b_3^2}}$$\cos 	heta = \frac{|2 	imes 4 + 2 	imes 1 + 1 	imes 8|}{\sqrt{2^2 + 2^2 + 1^2} \sqrt{4^2 + 1^2 + 8^2}}$$\cos 	heta = \frac{|8 + 2 + 8|}{\sqrt{4 + 4 + 1} \sqrt{16 + 1 + 64}}$$\cos 	heta = \frac{18}{\sqrt{9} \sqrt{81}} = \frac{18}{3 	imes 9} = \frac{18}{27} = \frac{2}{3}$Therefore,$	heta = \cos^{-1}(2/3)$.

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

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