Addition and Subtraction of Vectors
Figure 1, below, shows two vectors on a plane. To add the two vectors, translate one of the vectors so that the terminal point of one vector coincides with the starting point of the second vector and the sum is a vector whose starting point is the starting point of the first vector and the terminal point is the terminal point of the second vector as shown in figure 2.

When the components of the two vectors are known, the sum of two vectors is found by adding corresponding components.

Example 1
Given vectors A = (2 , -4) and B = (4 , 8), what are the components of

A → + B→

Solution

A → + B→ = (2 ,-4 ) + (4 , 8) = (2 + 4 ,-4 + 8 ) = (6 , 4)
The subtraction of two vectors is shown in figure 3. The idea is to change the subtraction into an addition as follows:

A → - B→ =
A → + (-B)→

Example 2

The magnitudes of two vectors U and V are equal to 5 and 8 respectively. Vector U makes an angle of 20° with the positive direction of the x-axis and vector V makes an angle of 80° with the positive direction of the x-axis. Both angles are measured counterclockwise. Find the magnitudes and directions of vectors U + V and U - V.

Solution

Let us first use the magnitudes and directions to find the components of vectors U and V.

Since both components of vector U + V are positive, the terminal side of angle θ is in quadrant I and therefore

θ = α = 60.9°

The direction of vector U + V is given by an angle approximately equal to 60.9°. This angle is measured in counterclockwise direction from the positive x-axis.

The signs of the components of vector U - V indicate that terminal side of angle β is in quadrant IV and therefore

β = 360° - α = 360° - 70° = 290°

The direction of vector U - V is given by an angle equal to 290°. This angle is measured in counterclockwise direction from the positive x-axis.

Example 3

The components of three vectors A, B and C are given as follows: A → = (2 , -1), B → = (-3 , 2) and C → = (13, - 8). Find real numbers a and b such that C → = a A → + b B →.

Solution

We first rewrite the equation C → = a A → + b B → using the components of the vectors.