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Find a vector `veca` of magnitude `5sqrt2` , making an angle of `π/4` with x-axis, `π/2` with y-axis and an acute angle θ with z-axis.

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#### Solution

Direction cosines of required vector a are

`"cos"^2 alpha + "cos"^2beta + "cos"^2 gamma = 1`

`"cos"^2 pi/4 + "cos"^2 pi/2 + "cos"^2 gamma = 1`

`(1/sqrt2)^2 + 0 + "cos"^2 gamma = 1`

`= 1/2 + 0 + "cos"^2 gamma = 1`

`"cos"^2 gamma = 1/2`

`"cos" gamma = 1/sqrt2`

`gamma = pi/4`

Vector for magnitude `5sqrt2`

`= 5sqrt 2 "cos" pi/4 + 5sqrt 2 "cos" pi/2 + 5sqrt 2 "cos" pi/4`

`= 5sqrt 2 1/sqrt 2 + 5(0) + 5sqrt 2 1/sqrt 2`

`veca =5hati+5hatk`

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