Addition and Subtraction of Vectors

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.

2 vectors vectors
Fig1. - 2 vectors in 2 dimensions.
addition of 2 vectors
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)→
subtract 2 vectors
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.

U → = (5 cos(20°) , 5 sin(20°))

V → = (10 cos(80°) , 10 sin(80°))

Magnitude and direction of vector U + V

U → + V→ = (5 cos(20°) , 5 sin(20°)) + (10 cos(80°) , 10 sin(80°))

= (5 cos(20°) + 10 cos(80°) , 5 sin(20°)+10 sin(80°))

Now that we have the components of vector U + V, we can calculate the magnitude as follows:

| U → + V→| = √ (5 cos(20°) + 10 cos(80°)) 2 + (5 sin(20°)+10 sin(80°)) 2 = 5√7 ≈ 13.22

If θ is the angle in standard position (angle between vector U+V and x-axis positive direction) of vector U + V, then

tan(θ) =
y-component of U+V / x-component of U+V
=
5 sin(20°)+10 sin(80°) / 5 cos(20°) + 10 cos(80°)


The reference angle α to angle θ is given by

α = arctan|(
5 sin(20°)+10 sin(80°) / 5 cos(20°) + 10 cos(80°)
)| ≈ 60.9°

We now approximate the components of vector U + V so that we can determine the quadrant of U + V

U → + V→ = (5 cos(20°) + 10 cos(80°) , 5 sin(20°)+10 sin(80°)) ≈ (6.43 , 11.6)

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.

Magnitude and direction of vector U - V

 U → - V→ = (5 cos(20°) , 5 sin(20°)) - (10 cos(80°) , 10 sin(80°))

= (5 cos(20°) - 10 cos(80°) , 5 sin(20°) - 10 sin(80°))

Now that we have the components of vector U - V, we can calculate the magnitude as follows:

| U → - V→| = √ (5 cos(20°) - 10 cos(80°)) 2 + (5 sin(20°) - 10 sin(80°)) 2 = 5√3 ≈ 8.66

If β is the angle in standard position (angle between vector U - V and x-axis positive direction) of vector U - V, then

tan(β) =
y-component of U-V / x-component of U-V
=
5 sin(20°) - 10 sin(80°) / 5 cos(20°) - 10 cos(80°)


The reference angle α to angle β is given by

α = arctan|(
5 sin(20°) - 10 sin(80°) / 5 cos(20°) - 10 cos(80°)
)| = 70°

We now approximate the components of vector U - V so that we can determine the quadrant of U - V

U → - V→ = (5 cos(20°) - 10 cos(80°) , 5 sin(20°) - 10 sin(80°)) ≈ (2.96 , -8.13 )

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.

(13, - 8) = a (2 , -1) + b (-3 , 2)

Rewrite an equation for each component

13 = 2 a - 3 b and - 8 = - a + 2 b

Solve the above equations in a and b to obtain

a = 2 and b = -3.

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