Motion in two dimensions, in a plane can be studied by expressing position, velocity and acceleration as vectors in Cartesian co-ordinates A = Axî + Ayĵ where î and ĵ are unit vector along x and y directions, respectively and Ax and Ay are corresponding components of (Fig. 4.9). Motion can also be studied by expressing vectors in circular polar co-ordinates as A = Ar
+ Aɵ 
where 
= 
= cosθ î + sin θ ĵ and 
= − sin θ î +cos θ ĵ are unit vectors along direction in which ‘r’ and ‘θ ’ are increasing.
(a) Express î and ĵ in terms of 
and 
.
(b) Show that both ř and θ are unit vectors and are perpendicular to each other.
(c) Show that 
(ř) = ω 
where 
and 
= −ωř
(d) For a particle moving along a spiral given by r= aθ 
, where a = 1 (unit), find dimensions of ‘a’.
(e) Find velocity and acceleration in polar vector representation for particle moving along spiral described in (d) above.
(a) We know,
(1)
(2)
Multiplying the equation (1) with sin
and (2) with cos
Adding the above two equations,
Substituting 
in equation (1),
(b) In order to show that 
and 
are perpendicular, lets show that their dot product is zero.
Therefore, 
and 
are perpendicular.
(c) 
Differentiating on both sides,
(d) 
Writing dimensions on both sides,
(e) We know,
Here, a=1
Differentiating on both sides to find velocity,
Differentiating on both sides to find acceleration,
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