0.26 Phy1230: vector subtraction  (Page 2/6)

My objective in this lesson is to explain vector subtraction in sufficient depth that you can visualize what it means when the text says that vector A issubtracted from vector B.

A simple rule

If A and B are scalars and you are asked to subtract A from B, you should already know the rule that says change the sign of A and then add.

The same rule also applies to vector subtraction. If we need to subtract vector Afrom vector B, we need to change the sign on the vector named A and then add it to the vector named B.

Changing the sign of a vector

So the question is, how do you change the sign of a vector? I will answer the question in three ways that really mean the same thing:

1. Change the sign for the horizontal and vertical components of the vector.
2. Draw the vector so that it points in exactly the opposite direction.
3. Add 180 degrees to the angle that defines the vector.

If all else fails, just remember this simple rule and apply it using one of the three ways described above.

Three ways to add vectors

Let's review three ways to add two vectors :

1. Draw them sequentially tail-to-head. The sum will be a vector that extends from the tail of the first vector to the head of the last vector.
2. Draw them tail-to-tail. Then draw a parallelogram where the two vectors form two sides of the parallelogram. The sum will be a vector that extendsfrom the point where the tails join to the opposite corner of the parallelogram.
3. Using trigonometry, decompose both vectors into horizontal and vertical components. Add the horizontal components and add thevertical components. The result will be the horizontal and vertical components of the sum vector. If needed, use trigonometry toconvert the components of the sum vector into a magnitude and angle.

Creating vectors with a graph board is useful

I have stated in earlier modules that the third approach using trigonometry is probably the most practical for blind students.

However, I also believe that it is important to get a picture in the mind's eye as to what happens when we add or subtract vectors.Therefore, I believe it is also useful for blind students to

• Use a graph board, a protractor, pipe cleaners, and pushpins to "draw" vectors.
• Add them using one of the first two approaches listed above .
• Develop a picture of the result in the mind's eye.

(Being able to see the addition and subtraction of vectors in the mind's eye is very important. As a sighted person, Ioften close my eyes and draw vectors on the palm of my hand in order to get a better feel for what happens when vectors are added or subtracted.)

Examples of vector subtraction

With that as an introduction, I am going to discuss some examples that are intended to help you to get a good feel for what it means to add, and moreimportantly to subtract two vectors. I hope that you will not only follow along and work through the examples, but that you will also use your graph board and "draw" the examplesas I describe them.

The parallelogram method

Before getting into the details of vector subtraction, let me make a few comments about the parallelogram method of vector addition.