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

Explanation

F.d = `F_(x)d_(x) + F_(y)d_(y)+F_(Z)d_(z)`
`=3(5) + 4(4)+(-5)(3)`
=16 unit
Hence, F.d= `Fdcostheta= 16 unit`
Now, F.F = `F^(2)= F_(x)^(2)+F_(Y)^(2)+F_(Z)^(2)`
`=9+16+25`
=50 unit
and d.d = `d^(2)= d_(x)^(2)+d_(y)^(2)+d_(z)^(2)`
=25+16+9
= 50 unit.
`therefore costheta= 16/(sqrt(50)sqrt(50)) = 16/50=0.32`
`theta=cos^(-1)` 0.32