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Question
Find the angle between the lines whose direction ratios are 4, –3, 5 and 3, 4, 5.
Solution
Let θ be the acute angle between the lines whose direction ratios are 4, –3, 5 and 3, 4, 5.
Then,
`costheta=|(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))|`
`cos theta=|(4(3)+(-3)(4)+5(5))/(sqrt(4^2+(3)^2+5^2).sqrt(3^2+4^2+5^2))|`
`=|(12-12+25)/(sqrt(16+9+25).sqrt(9+16+25))|`
`=|25/50|=1/2`
`costheta=1/2`
`theta=cos^-1(1/2)=pi/3`
The angle between the lines is `pi/3`
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