Find the angle between force `vec F = ( 5 hat i + 4 hat j + 5 hat k)` unit and displacement `vec d = (3 hat i + 4 hat j - 3 hat k)` unit. Also find the projection of `vec F on vec d.`

Explanation

`vec F. Vec d = F_(x)d_(x) + F_(y) d_(y) + F_(z) d_(z)`
`= 5(3) + 4(4) + 5(-3)`
`= 16 unit`
`F^(2) = F_(x)^(2) + F_(y)^(2) + F_(z)^(2)`
`= 5^(2) + 4^(2) + 5^(2)`
` = 66 unit`
`d^(2) = d_(x)^(2) + d_(y)^(2) + d_(z)^(2)`
` = 3^(2) + 4^(2) + (-3)^(2)`
`= 34 units`
`vec F. vec d = Fd cos theta`
`therefore cos theta = (vec F. vec d)/FD = 16/sqrt(66)sqrt(34) = 16/47.37 = 0.34`
`therefore theta = cos^(-1) (0.34)`
Projection of `vec F on vec d = F cos theta = sqrt66 xx 0.34 = 2.76`