Find the angle between force `vecF=(5hati+5hatj+5hatk)` unit and displacement `vecd=(3hati+4hatj-3hatk)` unit Also find the projection of `vecF on vecd.`

Explanation

`vecF.vecd=F_(x)d_(x)+F_(y)d_(y)+F_(z)d_(z)=5(3)+4(4)+5(-3)="16 unit"`
`vecF.vecF=F^(2)=F_(x)^(2)+F_(y)^(2)+F_(z)^(2)=5^(2)+4^(2)+5^(2)="66 unit"`
`vecd.vecd=d^(2)=d_(x)^(2)+d_(y)^(2)+d_(z)^(2)=3^(2)+4^(2)+(-3)^(2)="34 unit"`
`vecF. Vecd=Fd cos theta`
`therefore" "cos theta=(vecF.vecd)/(Fd)=(16)/(sqrt(66)sqrt(34))=(16)/(47.37)=0.34`
`therefore" "theta=cos^(-1)(0.34)`
`" Projection of "vecF" on "vecd=F cos theta =sqrt(66)xx0.34=2.76`