Vectors
Polar vs. Rectangular Form
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.
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 degrees is used from the 0-degree point. To use trigonometry, we need to determine what the angle is in reference to the X axis. In this example, it is 40 degrees (the supplement of 140 degrees). 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
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.
Rectangular to polar
Up to this point, we have been converting the polar form to rectangular. It is possible–and even important–to convert from rectangular to the polar form. If you have been given the rectangular coordinates, the best thing you can do is chart them out.
For instance, if you are given the coordinates 50, j80, then:
Step 1. Draw them out on an X-Y chart.
Step 2. Draw in the resultant and use Pythagoras’ theorem to determine its size.
Step 3. Use trigonometry to determine the angle.
Step 4. Using the resultant and the angle, go ahead and express it in polar form: 94 @ 58°
Congra