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Find the angle between the vectors with direction ratios proportional to `4,-3,5a n d3,4,5.`

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We have directions ratios of the lines as 4, -3,5 and 3 4,5.
Let `theta` be the required angle between the lines
`therefore " " cos = pm ((a_(1)a_(2)+b_(1)b_(2)+c_(1)c_(2))/(sqrt(a_(1)^(2) +b_(1)^(2)+c_(1)^(2)).sqrt(a_(2)^(2) +b_(2)^(2)+c_(2)^(2))))`
where , `a_(1) = 4, b_(1) = - 3, c_(1) = 5 and a_(2) = 3, b_(2) = 4, c_(2) = 5`
`therefore" " cos theta((4(3) - 3(4) +5(5))/(sqrt((4)^(2) +(-3)^(2)+(5)^(2))sqrt((3)^(2) +(4)^(2)+(5)^(2))))`
` = pm ((12-12+25)/(sqrt(16+9+25)sqrt(9+16+25)))`
` = pm ((25)/(50)) = pm (1)/(2)`
`therefore " "cos theta = pm (1)/(2)`
`therefore " " theta =cos^(-1) (pm (1)/(2))`
`rArr" "theta = (pi)/(3) or theta = (2pi)/(3)`

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