Consider the x - axis as representing east, the y - axis as north and z - axis as vertically upwards. Give the vector representing each of the following points and the direction is of 45°.
a) 5 m north east and 2 m up
b) 4 m south east and 3 m up
c) 2 m north west and 4 m up

Explanation

(a) Length along X-axis `=5cos45^(@)=(5)/(sqrt2)m`
Length along Y-axis `=5sin45^(@)=(5)/(sqrt2)m`
Length along Z-Axis =2m
`:.` In vector rotation `=(5)/(sqrt2)hati+(5)/(sqrt2)hatj+2hatk=(5(hati+hatj))/(sqrt2)+2hatk`
(b) Length along `X =4cos45^(@)=(4)/(sqrt2)m`
Length along `Y=4sin45^(@)=(4)/(sqrt2)m`
Length along Z-axis =3m
In vector rotation `=(4)/(sqrt2)hati-(4)/(sqrt2)hatj+3hatk=4(hati-hatj)//sqrt2+3hatk`
(c) Length along `X=-2cos45^(@)=-(-2)/(sqrt2)m=-sqrt2m`
Length along `Y=2sin45^(@)=(2)/(sqrt2)m=sqrt2m`
length along Z=4m
`:.` In vector rotation `=-sqrt2hati+sqrt2hatj+4hatk=(-i+j)sqrt2+4hatk`.