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Question
Find the angle between the following two lines:
`vecr = 2hati - 5hatj + hatk + λ(3hati + 2hatj + 6hatk)`
`vecr = 7hati - 6hatk + μ(hati + 2hatj + 2hatk)`
Solution
Angle between two vectors
`vecr = veca_1 + λvecb_1` and `vecr = veca_2 + μvecb_2` is given by
cos θ = `|(vecb_1 . vecb_2)/(|vecb_1||vecb_2|)|`
`vecr = (2hati - 5hatj + hatk) + λ(3hati + 2hatj + 6hatk)`
`veca_1 = 2hati - 5hatj + hatk, vecb_1 = 3hati + 2hatj + 6hatk`
`vecr = (7hati - 6hatk) + μ(hati + 2hatj + 2hatk)`
`veca_2 = (7hati - 6hatk), vecb_2 = hati + 2hatj + 2hatk`
Now `vecb_1 . vecb_2 = (3hati + 2hatj + 6hatk) . (hati + 2hatj + 2hatk)`
= 3 + 4 + 12
= 19
`|vecb_1| = sqrt(3^2 + 2^2 + 6^2)`
= `sqrt(49)`
= 7
`|vecb_2| = sqrt(1^2 + 2^2 + 2^2)`
= `sqrt(9)`
= 3
cos θ = `|b_1 . b_2|/(|b_1||b_2|)`
cos θ = `|19/(7 xx 3)|`
= `19/21`
θ = `cos^-1 19/21`
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