Conceptual Physics

3.5 - Adding and subtracting vectors graphically

Vectors can be added and subtracted. In this section, we show how to do these

operations graphically. For instance, consider the vectors A and B shown in Concept 1

to the right. The vector labeled A + B is the sum of these two vectors.

It may be helpful to imagine that these two vectors represent displacement. A person

walks along displacement vector A and then along displacement vector B. Her initial

point is the origin, and she would end up at the point at the end of the A + B vector.

The sum represents the displacement vector from her initial to final position.

To be more specific about the addition process: We start with two vectors, A and B,

both drawn starting at the origin (0, 0). To add them, we move the vector B so it starts

at the head of A. The diagram for Equation 1 shows how the B vector has been moved

so it starts at the head of A. The sum is a vector that starts at the tail of A and ends at

the head of B.

In summary, to add two vectors, you:

1. Place the tail of the second vector at the head of the first vector. (The order of
   addition does not matter, so you can place the tail of the first vector at the head of
   the second as well.)


2. Draw a vector between the tail end of the first vector and the head of the second
   vector. This vector represents the sum of two vectors.

To emphasize a point: You can think of this as combining a series of vector instructions.

If someone says, “Walk positive three in the x direction and then negative two in the y

direction,” you follow one instruction and then the other. This is the equivalent of placing

one vector’s tail at the head of the other. An arrow from where you started to where you

ended represents the resulting vector. Any vector is the vector sum of its rectangular

components.

When two vectors are parallel and pointing in the same direction, adding them is

relatively simple: You just combine the two arrows to form a longer arrow. If the vectors

are parallel but pointing in opposite directions, the result is a shorter arrow (three steps

forward plus two steps back equals one step forward).

To subtract two vectors, take the opposite of the vector that is being subtracted, and

then add. (The opposite or negative of a vector is a vector with the same magnitude but

opposite direction.) This is the same as scalar subtraction (for example 20 í 5 is the

same as 20 + (í5)). To draw the opposite of a vector, draw it with the same length but

the opposite direction. In other words, it starts at the same point but is rotated 180°. The diagram for Equation 2 shows the subtraction of two

vectors.

When a vector is added to its opposite, the result is the zero vector, which has zero magnitude and no direction. This is analogous to adding a

scalar number to its opposite, like adding +2 and í2 to get zero.

Adding vectors A + B graphically

Move tail of B to head of A

Draw vector from tail of A to head of B

Subtracting A íB graphically

Take the opposite of B

Move it to head of A

Draw vector from tail of A to head of íB

3.6 - Adding and subtracting vectors by components

You can combine vectors graphically, but it may be more precise to add up their

components.

You perform this operation intuitively outside physics. If you were a dancer or a

cheerleader, you would easily understand the following choreography: “Take two steps

forward, four steps to the right and one step back.” These are vector instructions. You

can add them to determine the overall result. If asked how far forward you are after this

dance move, you would say “one step,” which is two steps forward plus one step back.

You realize that your progress forward or back is unaffected by steps to the left or right.

You correctly process left/right and forward/back separately. If a physics-oriented dance

instructor asked you to describe the results of your “dancing vector” math, you would

say, “One step forward, four steps to the right.”

You have just learned the basics of vector addition, which is reasonably straightforward:

Break the vector into its components and add each component independently. In

physics though, you concern yourself with more than dance steps. You might want to

add the vector (20, í40, 60) to (10, 50, 10). Let’s assume the units for both vectors are

meters. As with the dance example, each component is added independently. You add

the first number in each set of parentheses: 20 plus 10 equals 30, so the sum along the

x axis is 30. Then you add í40 and 50 for a total of 10 along the y axis. The sum along the z axis is 60 plus 10, or 70. The vector sum is

(30, 10, 70) meters. If following all this in the text is hard, you can see another problem worked in Example 1 on the right.

Although we use displacement vectors in much of this discussion since they may be the most intuitive to understand, it is important to note that

all types of vectors can be added or subtracted. You can add two velocity vectors, two acceleration vectors, two force vectors and so on. As

illustrated in the example problem, where two velocity vectors are added, the process is identical for any type of vector.

Adding and subtracting vectors

by components

Add (or subtract) each component

separately

(^56) Copyright 2000-2007 Kinetic Books Co. Chapter 03