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

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