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

Explanation

Hint `vecF.vecd=F_(x)d_(x)+F_(y)d_(y)+F_(z)d_(z)`
`vecF.vecF=F^(2)=F_(x)^(2)+F_(y)^(2)+F_(z)^(2)" "vecd.vecd=d^(2)=d_(x)^(2)+d_(y)^(2)+d_(z)^(2)`
`cos theta =(vecF.vecd)/(Fd)`
`"Projection of "vecF" on "vecd=F cos theta`
`vecF.vecd=F_(x)d_(x)+F_(y)+d_(y)+F_(z)d_(z)=5(3)+4(4)+(-3)(5)="16 unit"`
`vecF.vecF=F^(2)=F_(x)^(2)+F_(y)^(2)+F_(z)^(2)=5^(2)+4^(2)+(-3)^(2)="50 unit"`
`vecd.vecd=d^(2)=d_(x)^(2)+d_(y)^(2)+x_(z)^(2)=3^(2)+2^(2)+5^(2)="50 unit"`
`cos theta=(vecF.vecd)/(Fd)=(16)/(sqrt(50)sqrt(50))=0.32 theta=cos^(-1)(0.32)`
`"Projection of "vecF" on "vecd=Fcos theta=sqrt(50)xx0.32=2.26`