In order to work out the actual direction in this example, we take 180 degrees and subtract 38.4 degrees from it to get 141.6 degrees. That means the direction must be somewhere between 90 and 180 degrees. We have to take into account that it is in the second quadrant. However, the polar form would not be 46 38.4 degrees. Using Pythagoras and trigonometry, we worked out that the resultant was 46 and the angle, in reference to the X axis, was 38.4 degrees. Confused? Look at the example below where the coordinates -36, j29 are given. When we work out the angle using the Pythagoras’ theorem, we use the angle in relation to the X axis but not the direction of the vector. This is especially important when dealing with the direction (angle). You need to be aware of what quadrant you are in. Remember that the X component is negative and the Y component is positive as they are in the second quadrant. If we are going to express it in rectangular form, use -46, j39. After that, we can use trigonometry to determine the X and Y components. In this example, it is 40 degrees (the supplement of 140 degrees). To use trigonometry, we need to determine what the angle is in reference to the X axis. The angle of 140 degrees is used from the 0-degree point. In the second quadrant, X is – (negative) and Y is + (positive). This puts the vector in the second quadrant. However, the letter i is also used as a symbol for current, so it was decided to go with the letter j instead. Because each is an imaginary number, the letter i was suggested. The X and Y components don’t really exist, and are referred to as imaginary numbers. The reason j is used is this.Īs 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 letter j is put in front of the y component to indicate the difference between the X and the Y. We then can express the same vector as 28.7, j 41. This is accomplished just by transposing the ratios from what we learned previously in trigonometry. We then can use the angle and the hypotenuse to determine the X axis with these equations: What does this look like to you? If you said right triangle, give yourself a pat on the back. ![]() Next, we draw a line straight down from the arrowhead to the X axis. ![]() The first step to finding this expression is using the 50 V as the hypotenuse and the direction as the angle. ![]() In the example below, we have a vector that, when expressed as polar, is 50 V 55 degrees. Rectangular form breaks a vector down into X and Y coordinates. It is more often the form that we like to express vectors in. Up to this point, we have used a magnitude and a direction such as 30 V 67°. When dealing with vectors, there are two ways of expressing them.
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