In this explainer, we will learn how to perform operations on vectors algebraically such as vector addition, vector subtraction, and scalar multiplication in two dimensions.
We are going to look at the and form of vectors; we will add and subtract vectors in this form and practice finding vector components given a magnitude and direction and vice versa.
First, let us recap the and vector format. The vector , as seen in the figure, is a unit vector, so it has length one, in the direction of the positive -axis.
The vector , as seen in the figure, is a unit vector in the direction of the positive -axis.
We can write the vector as and the vector similarly as
We can string together any number of and vectors. For example, the vector can be expressed using and notation. The vector takes us from to by traveling three units right and two units up. So is the same as . This can be seen in the figure.
We can add and subtract vectors in and form by treating the and components separately. For example, given a vector and a vector , the sum . We will demonstrate this in examples 1 and 2.
Example 1: Adding Vectors
Find the sum of the vectors and .
Firstly, let us draw a sketch of the two vectors.
If we write this out as a sum, we have
If we then group the horizontal and vertical components, we will be able to simplify the expression:
This process can be expressed visually by placing the vectors together, as seen in the figure, where the pink line represents the vector .
Example 2: Subtracting Vectors
Given the vectors and , calculate .
Firstly, we will write out the difference of the two vectors:
This time we need to be a little bit more careful as we are subtracting negatives. Remember, if you subtract a negative, this is equivalent to adding. So if we group the horizontal and vertical components as before, we have that
Simplifying, we find that
What we have done by subtracting is reverse its direction; this can be seen in the following figure.
If we are given a vector’s magnitude and its direction, we can convert this to find the vector’s component form using trigonometry. We will demonstrate this in the next couple of examples.
Example 3: Finding the Component Form of a Vector given Its Magnitude and Direction
The vector has a magnitude of 10 units and a direction of counterclockwise from the positive -axis. Write the vector in component form.
Let us start by drawing a sketch.
In order to convert to component form, we need to find its horizontal and vertical displacement, which can be calculated using trigonometry. First, it can be helpful to draw a right triangle and note the direction of each component and the value of the hypotenuse. Then, the component can be calculated by multiplying the magnitude (or hypotenuse) by and similarly the component can be calculated by multiplying the magnitude by , being the angle counterclockwise from the positive -axis.
In this example, the magnitude (or hypotenuse) is 10 and the value of is so the component is and the component is
Notice here that the the sign of the horizontal component is negative which reflects what we established when drawing our diagram. So, our vector can be written in component form as follows:
An alternative approach here would be to work out the acute angle in the right triangle and use standard right triangle trigonometry. If we do this, however, we need to be particularly careful in checking the direction of our vectors.
Example 4: Finding the Component Form of a Vector given Its Magnitude and Direction
The vector has a magnitude of 5 units and a direction of counterclockwise from the positive -axis. Write the vector in component form. Give the coefficients of and to two decimal places.
In order to answer this question, we need to first determine the quadrant in which the vector lies. We are told that the vector is positioned in a position above the positive real axis, which puts it in the first quadrant. If we then sketch out a right triangle to represent the vector, we get the following triangle.
Using standard right trigonometry, we can form an equation for . We have which gives
We can also form an equation for : which gives
Therefore, the component form of the vector is
We can generalize the method used in examples 3 and 4 to find the component form of a vector. Given a vector with magnitude and direction counterclockwise from the positive -axis, the component form of can be calculated using the formula
Now let us look at this process in reverse. If we are given a vector in component form, how do we find the vectors’ magnitude and direction? It is easier to approach this problem geometrically. We will explain this in example 5.
Example 5: Finding the Magnitude and Direction of a Vector in Component Form
Given the vector , find its magnitude and direction, giving the direction as an angle, , to two decimal places in the range .
Our first step in answering this question is to draw a diagram. This will help us identify the quadrant in which the vector lies.
We can see from our diagram that the vector lies in the fourth quadrant. We can calculate the magnitude, , of the vector by finding the length of the hypotenuse using the Pythagorean theorem:
We can work out the angle in the right triangle using trigonometry. Let be the angle of the vector below the -axis, if we consider the opposite to be and the adjacent to be 8, we have that
Notice here that is negative; the negative is informing us of a direction. A negative angle is considered to be measured in the clockwise direction and a positive angle is considered to be measured in the counterclockwise direction. Here, we have worked out the angle to in the clockwise direction from the positive -axis. However, the convention is to give a vector’s direction in a counterclockwise direction from the positive -axis. So, in our diagram will be equal to . In summary, our vector has a magnitude of 10 units and a direction of .
As before, we can generalize this result. For a vector , the magnitude will be equal to and the direction is calculated using the formula
With the direction, however, remember to be pay attention to the quadrant in which the vector lies, as the convention is to give the direction as an angle, counterclockwise from the positive -axis.
To convert vectors into component form, remember the following steps:
- It is often useful to draw a diagram of the vector to determine the quadrant in which it lies. You can then use standard trigonometry to determine the angle.
- Make sure you take note of the direction and reference point of the angle of the vector, for example, counterclockwise from the positive axis.
- You can use the general formula to find the component form of a vector:
Given a vector with magnitude and direction counterclockwise from the positive -axis, the component form of can be calculated using the formula
- The unit vector in the positive direction is and the unit vector in the positive direction is . Once the vectors are written in component form, you can easily find the sum or difference of two vectors by adding or subtracting the corresponding coefficients of the unit vectors.