Cartesian to Polar Form

Objective:

To convert vectors, complex numbers or coordinates from Cartesian form into Polar form.

General:

Cartesian coordinates, vectors and complex numbers are isomorphic systems, meaning they are all based on the same basis.
The x-coordinate, real part (of a complex number) or i-component (of a vector) can all be thought of as the horizontal component.
The y-coordinate, imaginary part (of a complex number) or j-component (of a vector) can all be thought of as the vertical component.
When converting, the forms (a,b), a+bi, ai+bj will be considered as equivalent.
tan θ° is determined by dividing the vertical component by the horizontal component.
Polar form will be written in the form r∠θ°, where r is the radius or distance from the origin and θ° is the angle from the positive x-axis in a counter-clockwise direction if positive and clockwise direction if negative.
Note that i in for the complex numbers is by definition √-1 and is sometimes represented by j. Hence we have i2=-1 or j2=-1. The i and j in the vector form are perpendicular unit vectors; i being in the positive x-direction and j in the positive y-direction. The vectors i and j have no direct relation to the complex number i or j and should not be confused.
In examples the complex i will be in italics and the vectors i and j will be bold faced.
All vectors will be represented in bold face.
For help with angles in all four quadrants, click here.

Method when the S and T scales are on the Slider:

Align the right index of S (equivalent to the right index of C) above the horizontal component on the D scale.
Move the cursor to the vertical component on the D scale.
Read the angle on the T scale.
Move the slider until this angle on the S scale is under the cursor.
Read the result below the C index.

Method when the S and T scales are on the Body:

Align the horizontal component on the C scale over the vertical component on the D scale.
Read the angle at the C index on the T scale.
Move the cursor to this angle on the S scale.
Align the vertical component on the C scale with the cursor.
Read the result above the D index.

Example 1: Convert the vector, v = 3i+2j into polar form. (With S and T scales on the slider.)

Align the right index of S (or C) with D3
Move the cursor to D2
Read the result T33.7
Align S33.7 with the cursor.
Read the result D3.6
Since tan θ° <1, the angle is less than 45° and the vector is in the first quadrant. Hence v = 3i+2j can be written as 3.6∠33.7°.

Example 2: Convert the complex number 5-6i into polar form. (With the S and T scales on the body.)

Align C5 with D6
Read T50.2 at the C index.
Move the cursor to S50.2
Align C6 with the cursor.
Read the result C7.8 at the D index.
Since tan θ° >1, the angle is greater than 45° and the number is in the fourth quadrant. Hence 5-6i can be written as 7.8∠-50.2°.

Practice Questions: Convert the following into polar form.