# Polar form and Cartesian form of Vector representation

Last updated on August 2nd, 2021 at 11:27 am

Here we will discuss the Polar form and Cartesian form of Vector representation. Vectors may be represented in two ways: Polar form and Cartesian form. Let’s discuss these in detail.

## Polar Form of Vector

In Polar form, a vector **A** is represented as **A** = (r, θ) where r is the magnitude and θ is the angle as shown in figure 1.

*vector resolv*ing using the polar form

Now let’s see how *vector resolv*ing is shown using the polar form. Let’s say, a vector **R** is resolved along X-axis and Y-axis. (2 dimensional). R_{x} and R_{y} are the resolved components along the X and Y-axis respectively. As shown in figure2, these two components are represented as follows:

R_{x}= R cosθ

R_{y}= R sinθ

These are also known as **rectangular components of a vector**.

## Cartesian form of Vector

In Cartesian form, a vector **A** is represented as ** A = **A

_{x}

**A**

**i**+_{y}

**A**

**j**+_{z}

**k**Here, A

_{x}, A

_{y}, and A

_{z}are the coefficients(magnitudes of the vector A along axes after resolution), and

**i, j**, and

**k**are the unit vectors along the X-axis, Y-axis, and Z-axis respectively. For example, A

_{x}is the magnitude of vector A resolved along the X-axis.

For simplification, let’s take the same vector **R** discussed under the Polar form of a vector, and let’s also consider a 2-D system (only X and Y-axis).

Following Cartesian form, **R= **R_{x}** i + **R_{y}** j**

## Mixing Polar & Cartesian

From the previous 2 sections,

Following Cartesian form, **R= **R_{x}** i + **R_{y}** j**

And, following Polar form,

R_{x}= R cosθ

R_{y}= R sinθ

So if we mix these 2 forms, we get: **R= **R cosθ** i + **R sinθ** j**

## Numerical solved – Polar to Cartesian conversion

1 ) Convert this vector presentation (10, 30**°**) of vector R to its cartesian form.

**solution: **If in polar form,

**R**= (10, 30

**°**)

To find out the cartesian form, we need to use the resolved or rectangular components of a vector.

then in cartesian form, **R** = 10 cos30** i** + 10 sin30** j**

=> **R** = 8.66** i** + 5** j**

## Numerical solved – Cartesian to Polar conversion

1 ) Convert this vector presentation **R** = 4** i** + 3** j** to its polar form.

**Solution**

We will use the Pythagoras theorem to get the polar form’s magnitude part.

|R| =√(4^{2} + 3^{2}) = 5

And, θ = tan^{-1} (3/5) = 31**°**