I was asked to show this video to students on how to set up their own vector calculator using Excel. The calculator gives you the ability to find the sum of vectors just by entering each vector's magnitude and direction.
Is it useful? Sure it is, but this is where if you don't know what is going on, you may be using it incorrectly! I have a couple of examples to show that.
After you have set up the calculator using the example shown in the video, enter these:
Vector A: Mag=9.8; Ref. Angle=114
Vector B: Mag=16.5; Ref. Angle= -104
Vector C: Mag=11; Ref. Angle=180
If you have set up the calculator correctly, you will get the resultant vector having a magnitude of 20.2 and a direction of 20.4 degrees.
If you don't know any better and I ask you to sketch out the direction of this vector, you would have drawn an arrow that is pointing in the first quadrant of a Cartesian coordinate system, which would be WRONG! In fact, most of my students would do that. It is a natural and automatic tendency to do so since angles are measured counter clockwise relative to the positive x-axis.
If you do a quick sketch and do a "tip-to-tail" vector addition, you will end up with a vector that is actually pointing in the 3rd quadrant! In fact, the true angular direction for this vector is 200.4 degrees (180 + 20.4, the latter is the angle found from the calculator).
The reason for this is that in calculating the angle, one is dividing the y-component by the x-component. This vector has both components being negative and so the division produces a positive value, producing a positive angle. But this angle given by the calculator, if one were to sketch out the vector, is the angle measured from the NEGATIVE x-axis, not the standard positive x-axis. If one remembers lessons from trigonometry, it is why the value of the tangent of an angle is positive in both the 1st and 3rd quadrant.
So the angle given is "correct" if one knows where it is measured from.
Here's another example to try:
Vector A: Mag=12.7; Ref. Angle=45
Vector B: Mag=19.2; Ref. Angle= -171
Entering this into the calculator, you get the resultant vector having a magnitude of 11.7 and angular direction of -30.9 degrees.
Once again, if you simply go by what you get, the tendency here is to think that the vector is in the 4th quadrant, because a negative angle means that it is an angle measured from the positive x-axis but going clockwise.
This is wrong.
The vector is actually in the 2nd quadrant. A simple sketch to do the vector addition will confirm that. The angle "-30.9 degrees" is actually an angle measured clockwise from the NEGATIVE x-axis. For this vector, the x-component of the resultant is negative, and thus, the ratio of the y-component to the x-component is negative, resulting in a negative value of the tangent and the angle. Once again, from trigonometric lesson, the tangent of an angle is negative in the 2nd and 4th quadrant.
What this all means is that a positive angle value is not unique - the vector could be in the 1st or 3rd quadrant - while a negative angle value is also not unique - the vector could be in the 2nd of 4th quadrant. Either do a quick sketch to do vector addition, or look at the sign of the resulting components.
There are two important lessons here. First is that one must know what the numbers mean. Using them blindly without understanding how they come about is risky and may result in the wrong conclusion. Secondly, for this exercise, there is no substitute for doing a sketch and knowing how vectors add. A simple sketch will provide an important sanity check that your conclusion about the vector direction is not wrong.
While this video and the setting up of the calculator is useful, the producer did not go far enough to demonstrate the possible pitfalls in using it blindly. There should have been examples involving what I had presented to tell the viewers what they should be careful about. I just wonder how many people had used this and interpreted their results incorrectly.
Zz.
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