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Wize University Physics Textbook (Master) > Force Vectors (Advanced Info)

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}

Practice: Night Bird Flying


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.
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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.

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Find the following, answering in cartesian vector form:
a) [2i+4j-6k] + [-3i-4j-10k]
b) [2i+4j-6k] - [-3i-4j-10k]