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.
Fig1. - 2 vectors in 2 dimensions.
Fig2. - Add 2 vectors in 2 dimensions - Parallelogram.
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)→
Fig3. - subtract 2 vectors.
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.