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

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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`