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Question

Find the angle between two lines whose direction ratio are
(i) $$2, 1, 2$$ and $$4, 8, 1$$         (ii) $$5, -12, 13$$ and $$-3, 4, 5$$


Solution

Angle between two lines whose direction ratios are:
$$a_1, b_1, c_1$$ and $$a_2, b_2, c_2$$ is
$$\cos\theta =\left|\dfrac{a_1a_2+b_1b_2+c_1c_2}{\sqrt{a^2_1+b^2_1+c^2_1}+\sqrt{a^2_2+b^2_2+c^2_2}}\right|$$
$$(1)$$ $$a_1=2, b_1=1, c_2=2$$;
$$a_2=4, b_2=8, c_2=1$$
$$\cos\theta =\dfrac{|8+8+2|}{\sqrt{4+1+4}\sqrt{6+64+1}}$$
$$\cos\theta =\dfrac{18}{3\times 9}$$
$$\cos\theta =\dfrac{2}{3}$$
$$\theta =\cos^{-1}\left(\dfrac{2}{3}\right)$$
$$(2)$$ $$a_1=5, b_2=-12$$, $$c_1=13$$, $$a_2=-3$$, $$b_2=4$$, $$c_2=5$$
$$\cos\theta =\dfrac{|-15-48+65|}{\sqrt{35+144+169}\sqrt{9+16+25}}$$
$$\cos\theta =\dfrac{|2|}{13\sqrt{2}5\sqrt{2}}$$
$$\cos\theta =\dfrac{1}{65}$$
$$\theta =\cos^{-1}\left(\dfrac{1}{65}\right)$$.

1116328_1157214_ans_1f0ed704756d4ce3aae9ab531aebdfb9.jpg

Mathematics

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