ax= ar sinΘ cosØ ax

ay =ar sinΘ sinØ ay

az=ar cosΘ

So, :

ax . ar =sin Θ cos Ø

ay.ar =sinΘ sin Ø

az.ar = cosΘ

___________________

ax =aΘ cosΘ cosØ

ay=aΘ cosΘ sinØ

az =aΘ (- sinΘ)

so,

ax.aΘ =cosΘ cosØ

ay.aΘ = cosΘ sinØ

az.aΘ = - sinΘ

__________________

ax=aØ (-sinØ)

ay=aØ cosØ

And the angle between Z ̄and a ̄Ø = zero

So,

ax.aØ = -sin Ø

ay.aØ = cosØ

az.aØ zero (since no component of Ø acting

in the direction of Z ̄

this depends on the study of Engineering

drawing and Descriptive geometry.

A.S

Θ

Z

X

Y

r

Ø

ar

aØ

aΘ

aΘ Ø

ax

ay