26 4.2 Vector Addition and Subtraction
Summary
- Define and apply the rules of vector addition and subtraction.
Vectors in One Dimension
A vector is a quantity that has magnitude and direction. Displacement, velocity, acceleration, and force, for example, are all vectors. In one-dimensional, or straight-line, motion, the direction of a vector can be given simply by a plus or minus sign. In the horizontal axis, + corresponds to movement to the right and – corresponds to movement to the left. In the vertical axis, + corresponds to upward movement and – corresponds to downward movement.
If all of the vectors are acting along the horizontal axis (x), vectors can be added or subtracted together like regular numbers. If all of the vectors are acting along the vertical axis (y), vectors can be added or subtracted together like regular numbers. For example, if a person runs 8 m to the right stops and then runs 10 m to the right, their final displacement is (+8m +10m) 18m. If the same person then walked 13 m to the left, the final displacement would be: +18m – 13 m = +5m.
If one of the vectors is acting along the horizontal axis (x) and one is acting along the vertical axis (y), you will have to use a different strategy, suitable for adding or subtracting vectors in two dimensions (x and y).
Vectors in Two Dimensions (One vector acting in y, one in x)
In two dimensions (2-d), we specify the direction of a vector relative to some reference frame (i.e., coordinate system), using an arrow having length proportional to the vector’s magnitude and pointing in the direction of the vector.
Figure 1 shows such a graphical representation of a vector, using as an example the total displacement for the person running in a city considered in the previous section. The person ran 9 blocks east and 5 blocks north and we want to know his final displacement. We shall use the notation that a symbol with an arrow over it, such as [latex]\vec{\text{D}}[/latex], stands for a vector. Its magnitude is represented by the symbol in italics, D, and its direction by θ.
VECTORS IN THIS TEXT
In this text, we will represent a vector with an arrow over a symbol. For example, we will represent the quantity force with the vector [latex]\vec{\text{F}}[/latex], which has both magnitude and direction. The magnitude of the vector will be represented by a variable in italics, such as F, and the direction of the variable will be given by an angle θ.
Vector Addition: Tip-to-Tail Method
The tip-to-tail method is a graphical way to add two vectors, when one is vertical and the other horizontal. It is described in Figure 2 below and in the steps following. The tail of the vector is the starting point of the vector, and the tip of a vector is the final, pointed end of the arrow.
Step 1. Draw an arrow to represent the first vector (9 blocks to the east) using a ruler.
Step 2. Now draw an arrow to represent the second vector (5 blocks to the north). Place the tail of the second vector at the tip of th