To add two or more vector quantities, we need a set of operations different from ordinary arithmetic. We can use three main methods to add two vectors: the head-to-tail method, the parallelogram method, and the addition by components.
Here, we will learn about these three methods for adding vectors. Also, we will look at some examples to apply the concepts.
Adding vectors with the head-to-tail method
Suppose an object has had a displacement of $latex \vec{A}$ followed by a second displacement of $latex \vec{B}$, as shown in the diagram below:
The final result is the same as if the object had started at the same starting point and had undergone a single displacement of $latex \vec{C}$.
The displacement $latex \vec{C}$ is the sum of the vectors $latex \vec{A}$ and $latex \vec{B}$ and is expressed symbolically as:
$latex \vec{C}=\vec{A}+\vec{B}$
The addition of two vectors is not the same operation as adding two scalar quantities, like 3+5=8.
To add two vectors with the head-to-tail method, we place the tail or base of the second vector at the head or tip of the first vector.
If we add the displacements $latex \vec{A}$ and $latex \vec{B}$ in reverse order, that is, $latex \vec{B}$ first and $latex \vec{A}$ second, the result is the same:
Then, we can observe that the order of the terms in an addition of vectors doesn't matter. That is, the commutative property applies in the addition of vectors
EXMPLE 1
Find the addition of the following three vectors:
Solution
To find the addition of the three vectors, we have to start by finding the sum of two of the vectors and then add the third vector to the result.
Then, we can start by adding the vectors $latex \vec{A}$ and $latex \vec{B}$ to obtain the vector $latex \vec{D}$:
Then, we add the vector $latex \vec{C}$ to the vector $latex \vec{D}$ to obtain the final result $latex \vec{R}$:
This addition can be done in any other order. For example, if we add $latex \vec{B}$ and $latex \vec{C}$ first and $latex \vec{A}$ last, we will obtain the same result.
EXAMPLE 2
A cyclist goes for a ride 10 km north and then 20 km east. How far and in which direction is she from the starting point?
Solution
We can start by drawing a diagram of the problem. Using the head-to-tail method, the addition of the vectors is:
Then, we want to find the magnitude and the direction (angle) of the dark blue vector, the vector $latex \vec{R}$.
Since the vectors form right angles to each other, the triangle formed is a right triangle. Then, we can use the Pythagorean theorem and trigonometry.
The distance from the starting point to the end point is equal to the length of the hypotenuse of the triangle:
$$\sqrt{(10\text{ km})^2+(20\text{ km})^2}=22.36\text{ km}$$
We can find the angle θ using the tangent function:
$$\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}=\frac{20\text{ km}}{10\text{ km}}=2$$
$$\theta=\tan^{-1}(2)=63.4^{\circ}$$
Then, the cyclist is 22.36 km away in a direction of 63.4° from North to East (positive x-axis).
Adding vectors with the parallelogram method
The parallelogram method is another way to represent an addition of two vectors graphically. Recall that a parallelogram is a quadrilateral in which its opposite sides are parallel.
Suppose we want to represent the following vector addition using the parallelogram method:
$latex \vec{C}=\vec{A}+\vec{B}$
We can draw the vectors $latex \vec{A}$ and $latex \vec{B}$ with their tails or bases at the same point. Then, we construct a parallelogram, where, $latex \vec{A}$ and $latex \vec{B}$ are two adjacent sides:
Then, the result of the addition, that is, the vector $latex \vec{C}$ is the diagonal of the constructed parallelogram.
EXAMPLE
Find the addition of the following vectors using the parallelogram method.
Solution
To find the resultant vector, we have to place the vectors with their bases at the same point. Then, we have:
Then, we form a parallelogram in which the initial vectors are its adjacent sides:
The addition of the vectors is equal to the diagonal of the parallelogram formed:
.
Adding vectors using their components
Two or more vectors can be added easily if we know their components. For this, we only have to add their components $latex x$ and $latex y$ separately.
Suppose we have two vectors $latex \vec{A}$ and $latex \vec{B}$ and we want to find the vector $latex \vec{C}$, which represents the addition of the two vectors.
We can use the following diagram to visualize this:
We can observe that the $latex x$ component of $latex \vec{C}$ is equal to the sum of the $latex x$ components of the vectors ($latex A_{x}+B_{x}$).
The same applies for the $latex and$ components. Thus, we have:
$latex C_{x}=A_{x}+B_{x}$
$latex C_{y}=A_{y}+B_{y}$
We can use this method to find the addition of any number of 2D or 3D vectors. For example, if $latex \vec{D}$ is the sum of $latex \vec{A}, \vec{B}, \vec{C}$, we have
$latex D_{x}=A_{x}+B_{x}+C_{x}$
$latex D_{y}=A_{y}+B_{y}+C_{y}$
EXAMPLE 1
Find the addition of the vectors $latex \vec{u}=3i+2j+5k$ and $latex \vec{v}=2i+j+3k$.
Solution
In this notation, the letters i, j, k represent the components in x, y, and z respectively.
Then, to find the components of the vector formed by the addition of $latex \vec{u}$ and $latex \vec{v}$, we add the components of the vectors:
$$R_{x}=u_{x}+v_{x}=3+2=5$$
$$R_{y}=u_{y}+v_{y}=2+1=3$$
$$R_{z}=u_{z}+v_{z}=5+3=8$$
Therefore, the result of the addition of the vectors is
$latex \vec{R}=5i+3j+8k$
EXAMPLE 2
Find the components of the vector formed by the addition of the following vectors:
$latex \vec{A}$: 20 m, 60 ° from East to North
$latex \vec{B}$: 10 m, 30 ° from East to North
Solution
To solve this problem, we have to start by finding the components in $latex x$ and in $latex y$ of the two given vectors.
Then, we use the formulas of the components of a vector, remembering that we find the $latex x$ component with the cosine and the $latex y$ component with the sine:
$$A_{x}=A\cos (\theta)=(20\text{ m})(\cos(60^{\circ})=10\text{ m}$$
$$A_{y}=A\sin (\theta)=(20\text{ m})(\sin(60^{\circ})=17.32\text{ m}$$
$$B_{x}=B\cos (\theta)=(10\text{ m})(\cos(30^{\circ})=8.66\text{ m}$$
$$B_{y}=B\sin (\theta)=(10\text{ m})(\sin(30^{\circ})=5\text{ m}$$
Therefore, the components of the resultant vector are:
$$R_{x}=A_{x}+B_{x}=10\text{ m}+8.66\text{ m}=18.66\text{ m}$$
$$R_{y}=A_{y}+B_{y}=17.32\text{ m}+5\text{ m}=22.32\text{ m}$$
See also
Interested in learning more about vectors? Check out these pages:
