2.3: Polar vs. Rectangular Form

Last updated
Save as PDF

Page ID: 2639

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

Polar form

When dealing with vectors, there are two ways of expressing them. Up to this point, we have used a magnitude and a direction such as 30 V @ 67°. This is what is known as the polar form. It is more often the form that we like to express vectors in.

Rectangular form

Rectangular form breaks a vector down into X and Y coordinates. In the example below, we have a vector that, when expressed as polar, is 50 V @ 55 degrees. The first step to finding this expression is using the 50 V as the hypotenuse and the direction as the angle. Next, we draw a line straight down from the arrowhead to the X axis. What does this look like to you? If you said right triangle, give yourself a pat on the back. We then can use the angle and the hypotenuse to determine the X axis with these equations:

cos 55°×50 = 28.7 for the X axis
sin 55°×50 = 41 for the Y axis

This is accomplished just by transposing the ratios from what we learned previously in trigonometry.

Figure 31. Quadrant 1

We then can express the same vector as 28.7, j 41.

Where did that j come from?

The letter j is put in front of the y component to indicate the difference between the X and the Y. The reason j is used is this.

As a way of telling the difference between X and Y, it was decided that a letter should be put in front of the Y. The X and Y components don’t really exist, and are referred to as imaginary numbers. Because each is an imaginary number, the letter i was suggested. However, the letter i is also used as a symbol for current, so it was decided to go with the letter j instead.

Why polarity is important

Let’s look at another example. The polar form is 60 V @ 140 degrees. This puts the vector in the second quadrant.

In the second quadrant, X is – (negative) and Y is + (positive). The angle of 140° is used from the 0° point. To use trigonometry, we need to determine what the angle is in reference to the X axis. In this example, it is 40° (the supplement of 140°). After that, we can use trigonometry to determine the X and Y components.

cos 40°×60 = 46 for the X axis
sin 40°×60 = 39 for the Y axis

Figure 32. Quadrant 2

If we are going to express it in rectangular form, use -46, j39. Remember that the X component is negative and the Y component is positive as they are in the second quadrant.

Video! This video walks through how to convert from polar form to rectangular form.