A cube is placed so that one corner is at the origin and three edges are along the x-,y-, and ,z-axes of a coordinate system (figure).Use vector to compute
a.The angle between the edge along the z-axis (line ab) and the diagonal from the origin to the opposite corner (line ad).
b. The angle between line ac (the diagonal of a face ) and line ad.

Answer

a.`cos^(-1)(1/sqrt(3))`; b. cos^(-1)(sqrt(2)/sqrt(3))

Explanation

a.Let side of the cube is `d`,then `vec(ab)=dhat(k)` and `vec(ad)=dhat(i)+dhat(j)+dhat(k)`
`cos theta=(vec(ab).vec(ad))/(|vec(ab)||vec(ad)|)=(d^(2))/(dsqrt(3d))=1/sqrt(3)rArr theta =cos^(-1)(1/sqrt(3))`
`vec(ac)=dhat(j)+dhat(k)`
`cos theta=(vec(ac).vec(ad))/(|vec(ac)||vec(ad)|)=(2d^(2))/sqrt(2dsqrt(3d))=sqrt(2)/sqrt(3)`
`rArrcos^(-1)((sqrt(2))/sqrt(2))`