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Wize University Physics Textbook (Master) > Introduction to Vectors

Vector Addition and Subtraction

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Vector Addition and Subtraction: Algebraic Method


In general, we can perform addition and subtraction operations on vectors either algebraically or graphically.

Algebraic Method

If we know the components of vectors, we can add/subtract them algebraically by adding/subtracting their corresponding components:

v+u=[v1v2]+[u1u2]=[v1+u1v2+u2]v+u=\begin{bmatrix}v_1\\v_2\end{bmatrix}+\begin{bmatrix}u_1\\u_2\end{bmatrix}=\begin{bmatrix}v_1+u_1\\v_2+u_2 \end{bmatrix}

v−u=[v1v2]−[u1u2]=[v1−u1v2−u2]v-u=\begin{bmatrix}v_1\\v_2\end{bmatrix}-\begin{bmatrix}u_1\\u_2\end{bmatrix}=\begin{bmatrix}v_1-u_1\\v_2-u_2 \end{bmatrix}


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Vector Addition and Subtraction: Graphical Methods

We can add vectors using two different graphic methods described below:





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Subtraction

Now what about subtraction? Say, I subtracted B⃗\vec{B} from A⃗\vec{A} . Well that would be the same as saying, add the negative of B⃗\vec{B} .
A⃗−B⃗=A⃗+(−B⃗)=C⃗\vec{A} - \vec{B} = \vec{A }+ ( - \vec{B}) = \vec{C}

What is −B⃗-\vec{B} graphically? Well it should cancel B⃗\vec{B} when we add them together. So it must be exact same vector but flipped around.



Alternatively, we can use the parallelogram method to find the subtraction of two vectors as shown in the picture.

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Example: Finding Missing Corner of a Parallelogram


The coordinates of four corners of a Parallelogram ABCDABCD in a Clock-Wise order are A=(1,3)A=(1,3), B=(−1,5)B=(-1,5), CC and D=(4,7)D=(4,7). Find missing coordinates of CC.

Solution:

We can start by drawing a picture using the informations already given to us and use the properties of the vectors to find a vector which is related to the missing point.

AC→=AB→+AD→;\overrightarrow{AC} = \overrightarrow{AB}+\overrightarrow{AD};

AB→=B−A=(−1,5)−(1,3)=(−2,2)\overrightarrow{AB} = B-A = (-1,5)-(1,3)=(-2,2)

AD→=D−A=(4,7)−(1,3)=(3,4)\overrightarrow{AD} = D-A = (4,7)-(1,3)=(3,4)

AC→=(−2,2)+(3,4)=(1,6)\overrightarrow{AC} = (-2,2)+(3,4) = (1,6)

AC→=C−A→C=AC→+A\overrightarrow{AC} = C-A \to C=\overrightarrow{AC}+A

C=(1,6)+(1,3)=(2,9)→C CoordinateC=(1,6)+(1,3)=(2,9) \to C\ \text{Coordinate}
In the diagram shown, the unknown vector is


checklist
Mark Yourself Question
  1. Grab a piece of paper and try this problem yourself.
  2. When you're done, check the "I have answered this question" box below.
  3. View the solution and report whether you got it right or wrong.
Two forces F1F_1 and F2F_2 are shown in the figure. Find the magnitude and direction of the resultant using law of vector addition.

Extra Practice
If you walked 3.00km West, and then 5.00km South to get to the store, what angle are you traveling at relative to the South direction? What is the magnitude of your displacement?
If you walked 3.00km West, and then 5.00km South to get to the store, what angle are you traveling at relative to the South direction? What is the magnitude of your displacement?
Practice: Resultant of 2 Vectors
Two forces F1F_1 and F2F_2 are shown in the figure. Find the magnitude and direction of the resultant using law of vector addition.
Find the missing vector
In the diagram shown, the unknown vector is
Resultant of 2 Vectors
Two forces F1F_1 and F2F_2 are shown in the figure. Find the magnitude and direction of the resultant using law of vector addition.
Practice: Vector Addition and Subtraction
The coordinates of four corners of a Parallelogram ABCDABCD in a Clock-Wise order are A=(1,3)A=(1,3), B=(−1,5)B=(-1,5), CC and D=(4,7)D=(4,7). Find missing coordinates of CC.
Given an object that is moved 1 m1\ mWest then 3 m3\ mNorth then 2 m2\ mEast and finally 1 m1\ mNorth.
a) What is the final displacement of the object?
b) What is to total distance traveled?
Equilibrium: Static
Three forces act on a ball in different directions: 100N at 32°, 72 N at 135°, and 125 N at 294°. Find the fourth force required to balance these forces.
If you walked 3.00km West, and then 5.00km South to get to the store, what angle are you traveling at relative to the South direction? What is the magnitude of your displacement?
A bird flew 100m to the east then 200m to the northwest. Use the algebraic addition of vectors to find the bird’s net displacement.
Subtract vectors algebraically
Add the following vectors:
A⃗ = 2.0i^−3.0j^\vec{A}\ =\ 2.0\hat{i} - 3.0\hat{j}
B⃗\vec{B} which has a magnitude of 4.0 at 23o left of the -y axis.
Sum of vectors
What would be the resultant vector from the addition of the below three vectors?
In the diagram shown, the unknown vector is
What is the resultant of a pair of forces, 100 N upward and 75 N downward?
Enter answer in N. positive is upward and negative is downwards.
What is the magnitude of the resultant of two vectors of magnitude 4 and 3 that are at right angles to each other?
Practice: Resultant of 2 Vectors
Two forces F1F_1 and F2F_2 are shown in the figure. Find the magnitude and direction of the resultant using law of vector addition.
Practice: Negative of a Vector
R⃗\vec{R} is shown in the figure. Draw −R⃗-\vec{R}
If vector A is A = (2,−2,2)A\ =\ \left(2,-2,2\right) and vector B is B = (3,4,5)B\ =\ \left(3,4,5\right) find the length of A+BA+B.
Given an object that is moved 1 m1\ m West then 3 m3\ m North then 2 m2\ m East and finally 1 m1\ m North.
a) What is the final displacement of the object?
b) What is to total distance traveled?
If vector A = (1,2,3)A\ =\ \left(1,2,3\right) and vector B = (−7,5,4)B\ =\ \left(-7,5,4\right) find A+BA+B.
Practice: Vector Addition and Subtraction
The coordinates of four corners of a Parallelogram ABCDABCD in a Clock-Wise order are A=(1,3)A=(1,3), B=(−1,5)B=(-1,5), CC and D=(4,7)D=(4,7). Find missing coordinates of CC.