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

Explanation

`vec(F) =3hat(i)+4hat(j)-5hat(k)`
`vec(d) =5hat(i)+4hat(j)+3hat(k)`
`vec(F).vec(d) =(3hat(i)+4hat(j)-5hat(k)).(5hat(i)+4hat(j)+3hat(k))`
` = 15+16 - 15`
= 16 unit
` | vec(F) | = sqrt((3)^(2)+(4)^(2)+(-5)^(2))`
`= sqrt(9+16+25)`
` =sqrt(50)`
`|vec(F)| =sqrt((3)^(2)+(4)^(2)+(-5)^(2))`
`=sqrt(25+16+9)`
`= sqrt(50)`
`|vec(d)|=sqrt((3)^(2)+(4)^(2)+(-5)^(2))`
Now `vec(F).vec(d) =| vec(F) | |vec(d)| cos theta `
` :. cos theta= (vec(F).vec(d))/(|vec(F)||vec(d)|)`
`= 16/((sqrt(15))(sqrt(50)))`
` =16/15`
` :. cos theta =0.32`
`sin(90^(@) - theta) = 0.32 `
`90^(@) - theta =18^(@)40`
` :. theta = 90^(@) - 18^(@) 40`
` :. theta = 71^(@) 21 `
Projection of F on d
=`F cos theta `
` = sqrt(50) xx0.32 `
` = 2.263` N