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Question
Find the acute angle between the following lines.
`(x + 4)/3 = (y - 7)/4 = (z + 5)/5, vec"r" = 4hat"k" + "t"(2hat"i" + hat"j" + hat"k")`
Solution
`(x + 4)/3 = (y - 7)/4 = (z + 5)/5, vec"r" = 4hat"k" + "t"(2hat"i" + hat"j" + hat"k")`
`vec"b" = 3hat"i" + 4hat"j" + 5hat"k"`
`vec"d" = 2hat"i" + hat"j" + hat"k"`
`vec"b" * vec"d" = 3(2) + 4(1) + 5(1)`
= 6 + 4 + 5
= 15
`|vec"b"| = sqrt(9 + 16 + 25)`
= `sqrt(50)`
= `5sqrt(2)`
`|vec"d"| = sqrt(4 + 1 + 1)`
= `sqrt(6)`
cos θ = `|vec"b" * vec"d"|/(|vec"b"||vec"d"|)`
= `15/(5sqrt(2) xx sqrt(6))`
= `3/(sqrt(2)*sqrt(2)sqrt(3)`
= `3/(2sqrt(3))`
= `sqrt(3)/2`
θ = `pi/6`
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