The axes of coordinates are rotated about the z-axis though an angle of `pi//4` in the anticlockwise direction and the components of a vector are 2`sqrt(2),` 3`sqrt(2), 4.` Prove that the components of the same vector in the original system are -1,5,4.

Explanation

If `hati, hatj, hatk` are the new unit vectors along the coordinate axes, then
`" " veca = 2sqrt2 hati + 2sqrt2hatj + 4hatk" "` (i)
`hati, hatj, hatk` are obtained by rotating by `45^(@)` about the z-axis.
Then `hati` is replaced by `hati cos 45^(@) + hatj sin 45^(@) = (hati + hatj)/(sqrt2)`
and `hatj` is replaced by
`" "-haticos45^(@)+hatj sin 45^(@) = (-hati + hatj)/(sqrt2)`
` " " hatk = hatk`,
`" " veca = 2sqrt2 [(hati +hatj)/(sqrt2)] +3sqrt2[(-hati +hatj)/(sqrt2)] + 4hatk`
`" " = (2-3)hati + (2+3)hatj + 4hatk`
`" " = - hati +5hatj + 4hatk`