In this explainer, we will learn how to show that motion in directions that are at right angles to each other can be represented by motion in one direction.

A full description of certain physical quantities such as displacement, velocity,
acceleration, and force must include information about **direction** as well as
**magnitude**. As a result, we generally represent these quantities using
**vectors**, mathematical objects that have both **magnitude** and
**direction**. Using vectors to represent vector quantities enables us to add
these quantities together in a meaningful way as well as perform many other useful
mathematical functions with them.

Recall that vectors can be added together using a graphical method called the
**tip-to-tail** method. It works like this: to add two vectors
(represented by arrows) together, move one of the vector arrows so that its
*tail* touches the *tip* of the other vector. Then, draw a new vector
from the tail of the unmoved vector to the tip of the moved vector. This new
vector is the sum of the two original vectors.

For example, we can add together the blue and yellow vectors shown on this grid.

First we move the orange vector so that its tail touches the tip of the blue vector.

Now, we will draw a new vector in pink, going from the *tail* of the
unmoved (blue) vector to the *tip* of the moved (orange) vector.

This pink vector is equal to the sum of the blue vector and the orange vector.
We can equivalently say that it is the **resultant** of these two vectors.

We can express the relationship between the blue, orange, and pink vectors
algebraically using **vector notation**. Letβs call the blue vector
, the orange vector
, and the pink vector
.

Since we have shown that the pink vector is the sum of the other two vectors, we can write

We would also get the same result if we used the tip-to-tail method to add the vectors in the opposite order:

So, we have shown that if we add together a vector with a magnitude of 3 pointing to
the right and a vector with a
magnitude of 5 pointing upward ,
we get a new vector. This new vector, which we have called
, points at an angle (it goes up *and* right),
and its **magnitude** (i.e., its length) appears to be longer than either
or , but not
quite as long as both of them put together. The fact that the magnitude of this
vector is not simply the sum of the magnitudes of and
shows us that vectors cannot just be added together like normal
numbers.

The central idea of this explainer is that **vector quantities** in physics
obey the rules of vector addition. In other words, vector quantities of the same
typeβforces, for exampleβcan be added together in exactly the same
way as the arrows on our graph.

This means that if we have a
3 N force pushing an object to
the right (which we could represent with vector )
and a 5 N force pushing the
same object upward (which we could represent with the vector
), the combined force that acts on the object
(in other words, the **resultant** force) is represented by vector
.

Next, letβs look at how we can add vector quantities in an accurate way
*without* drawing them.

Adding vectors accurately depends on the idea of **components**. Recall that we
use components to talk about vectors in terms of how far they go in the horizontal
direction and how far they go in the vertical direction. Letβs consider two
new vectors, and
.

We can express these vectors as the sums of their components:

Describing vector quantities in terms of their components in the horizontal and vertical directions makes it easy to add them together. If we want to add together vectors and , we just need to add together their horizontal components and their vertical components separately:

We have found that the **resultant** vector created by adding
and has a
horizontal component of 1 and a vertical component of 3.

Letβs call this resultant vector and add it to our diagram.

The same relationship holds for vector quantities in physics. For example, a body subjected to a force represented by and another force represented by would experience a resultant force corresponding to .

### How To: Adding Vectors Together Using Their Components

To add together two vectors and ,

- find the horizontal and vertical components of and ,
- note that the horizontal component of is the sum of the horizontal components of and ,
- note that the vertical component of is the sum of the vertical components of and .

We have now seen two different ways of expressing vectors:

**Component form**: we can express a vector as the sum of its horizontal and vertical components.**Magnitude and direction form**: we can express a vector in terms of its overall magnitude and direction.

We have seen that **component form** is useful for adding vectors together.
However, it is often more useful and intuitive to express vector quantities in
terms of their **magnitude and direction**. Being able to convert between these
two forms is the final step in effectively using vectors in physics. Letβs
start with converting from **component form** to **magnitude and direction
form**.

Letβs consider an object whose velocity in metres per second (m/s) is represented by vector . Letβs say that has a horizontal component of 4 and a vertical component of .

What is the magnitude and direction of the objectβs velocity?

We can calculate both of these using geometry. First, letβs draw the horizontal and vertical components and onto our diagram.

Here, we can remember that the vector is equivalent to the sum of its components. That is,

Remember that in this equation, , , and are all vectors. This means that we cannot simply add the magnitudes of and to get the magnitude of .

Notice that and its components form a
**right triangle**. This enables us to use geometry to calculate the magnitude
and direction of the objectβs velocity. First, we can calculate the
magnitude (i.e., the length) of ,
, using the
**Pythagorean theorem**. Consider a right triangle with a hypotenuse of length
and two other sides of lengths
and , as shown in the diagram below.

The lengths are related by the equation

Applying this to the triangle formed by the vector and its components, we have where and are the magnitudes (lengths) of the horizontal and vertical components respectively. Substituting these lengths into our equation, we have

So, the magnitude of the objectβs velocityβthat is, its speedβis 5 m/s.

### Equation: Finding the Magnitude of a 2D Vector Based on its Components

The magnitude of any 2-dimensional vector is given by where and are the horizontal and vertical components of respectively.

Letβs look at an example where we put this idea into practice.

### Example 1: Finding the Magnitude of Two Perpendicular Displacement Vectors

Point is located 8 m horizontally from the base of the wall of a house, and point is located 6 m vertically above the base of the wall, as shown in the diagram. What is the magnitude of the displacement from point to point ?

### Answer

The key to answering this question is to think about the
*displacement vector* from to
. We can call this vector
. It describes the magnitude and direction
of the displacement of point Β *from*
(or *relative to*) point .

Notice that in this case, the horizontal and vertical components of are labeled in the diagram.

Following the usual convention of defining our horizontal direction as pointing to the right and our vertical direction as pointing up, we can express as the sum of its componenets as

We can calculate the **magnitude** of any vector using its components.
The magnitude of any 2-dimensional vector
is given by
where and
are the horizontal and vertical
components of respectively.

In this question, we want to find the magnitude of the vector . Substituting its components into the equation above, we have

We can then evaluate the magnitudes on the right-hand side (i.e., ignore any negative signs) and then work out the magnitude as follows:

Because the lengths we used were expressed in metres, our answer is expressed in metres too. So we obtain a final value for the magnitude of the displacement from to of 10 m.

Letβs get back to our velocity vector . Now that we have found its magnitude, letβs find its direction. We can do this using trigonometry.

Recall that for any right triangle, the angle , the length of the side opposite that angle O, and the length of the side adjacent to that angle A are related by the equation

We can apply this equation to our vector .

Note that the horizontal component is adjacent to the angle and the vertical component of is opposite it. This gives us the equation

To find the angle of our velocity vector below the horizontal, we just need to rearrange the equation to make theta the subject:

Substituting the magnitudes (lengths) of and into the equation gives us

So, we know that points at an angle of below the horizontal axis.

Letβs look at an example question where we need to calculate the direction of a vector based on its components.

### Example 2: Finding the Direction of a Displacement Vector

A bird flies along a line that displaces it 450 m east and 350 m north of its starting point, as shown in the diagram. What angle must the bird turn toward the west to change direction and fly directly north? Give your answer to the nearest degree.

### Answer

In the diagram, the blue arrow represents the birdβs
**displacement vector**. The black arrows, pointing
450 m to the east and
350 m to the north,
effectively act as horizontal and vertical axes respectively. These
measurements tell us that the horizontal component of the birdβs
displacement is 450 m to
the east, and the vertical component of the birdβs displacement is
350 m to the north.

Letβs start by modifying our diagram to show the angle through which the bird would have to turn in order to point north.

This is the angle that we are trying to find. Notice that this is the same as the angle of the birdβs displacement vector from the vertical (north) axis.

This means we are effectively looking to find the *angle at which the bird
was flying*, as measured clockwise from the vertical axis. We could
equivalently say that we are looking for the direction of the birdβs
**displacement vector**, measured clockwise from the vertical axis.

We can work out the direction in which a vector points using trigonometry. First, we arrange the birdβs displacement vector and its vertical and horizontal components to form a right triangle.

The length of the hypotenuse is the magnitude of the birdβs displacement vector, the length of the side adjacent to is the magnitude of the vertical component of the displacement vector, and the length of the side opposite is the magnitude of the horizontal component of the displacement vector.

Letβs label these sides H, A, and O respectively.

We know O and A, and we need to work out the angle . We can do this using the following trigonometric relationship:

To make the subject, we take the inverse tangent of both sides:

Now, we just need to substitute in the lengths O and A:

So, the bird needs to turn this far in the counterclockwise direction and it will face north. All we need to do now is round this to nearest degree to get our final answer, which is .

Now, letβs see how we can do things the other way around. We can calculate
the *components* of a vector if we are *given* its magnitude and
direction. Consider a force vector with magnitude
8 N acting at an angle of
above the horizontal to
the left.

Letβs show this vector on a grid along with its horizontal and vertical components.

Now, we can see that the horizontal component of , , is negative, and it appears to have a magnitude just less than 7. The vertical component, , is positive, and it looks like it has a magnitude of 4 units.

To find the exact lengths of these components, we can use trigonometry.

Recall that for any right triangle, the angle , the length of the side opposite that angle O, the length of the side adjacent to that angle A, and the length of the hypotenuse H are related by two trigonometric equations:

We can apply these formulas to our force vector, , to determine the magnitudes of its components. In our diagram, , , and . Substituting these quantities into our trigonometric equations, we have

Multiplying both sides of each equation by the magnitude of gives us

All we need to do now is substitute in the values and to obtain the magnitudes of the - and -components of :

We have now found that the magnitude of the vertical component of is 4, and the magnitude of the horizontal component of is (which, expressed as a decimal and rounded to 2 decimal places, is 6.93).

Looking back at the previous diagram, we can see that the vertical component of
points in the *positive*
vertical direction, while the horizontal component of
points in the *negative* horizontal direction. When we express
as the sum of its components, we make this clear by
our use of positive and negative signs:

In other words, the 8 N
force represented by the vector is equivalent to
a force of
N
acting to the left *and* a
4 N force acting upward.
Converting vectors into their component form is essential if we want to be able to
add vectors together.

### Example 3: Finding the Components of a Displacement Vector

A surveyor walks through a field, as shown in the diagram. How much further east does the surveyor walk than he walks north? Round your answer to the nearest metre.

### Answer

In the diagram, the surveyorβs displacement vector is represented by the red arrow. We can see that he walks 450 m at an angle of , measured counterclockiwse from the east direction.

The question asks how much further the surveyor walks east than he walks north. In order to answer this, we can work out the components of displacement in the east and north directions.

Letβs start by considering the right triangle formed by the displacement vector and its components. We will call the surveyorβs displacement vector , and we will call its vertical (north) and horizontal (east) components and respectively.

For a right triangle of sides A and O and a hypotenuse H, if is the angle between the side A and the hypotenuse H, then the length of A can be given by

In both cases, H is the length of the hypotenuse.

In this case, H is the magnitude of the surveyorβs displacement vector (450 m), A is the magnitude of the horizontal component of this vector, and O is the vertical component of this displacement vector. We also know that the angle, , is . So,

So, we can see that the horizontal component (i.e., displacement in the east direction) is larger than vertical component (i.e., displacement in the north direction). To find out exactly how much further the surveyor walked to the east than to the north, we just need to find the difference between these two components:

Rounding this to the nearest metre, we have a final answer of 165 m.

Now, letβs look at an example where we need to add together two vectors
acting at different angles. We first need to convert these vectors into component
form so we can add them, then we convert the resultant vector *back* into
magnitude and direction form.

### Example 4: Finding the Resultant of Two Forces Acting at Different Angles

The force is the resultant of the two force vectors shown in the diagram. What is the magnitude of to the nearest newton?

### Answer

In the diagram, we can see that the two red vectors have been arranged so that they are tip to tail. The vector spans from the tail of one vector to the tip of the other. This is a graphical way of showing that the vector is equal to the sum (or the βresultantβ) of the two red vectors.

If we call the 70 N force vector and the 60 N force vector , then we can express this relationship as an equation:

In order to find the magnitude of , we need to add and together. To do this, we will have to first express and in component form.

Letβs start with . We can draw a diagram where and its horizontal and vertical components (represented by the blue and pink arrows respectively) form a right triangle.

We can use trigonometry to tell us the magnitudes of the side opposite the angle, O, and the side adjacent to the angle, A, in terms of the magnitude of the hypotenuse H and the angle :

We know that the magnitude of the hypotenuse, H, is 70 N and the angle is :

These βside lengthsβ are the magnitudes of our horizontal and vertical components expressed in newtons. Rounding these to two decimal places, we can express the 70 N force as the sum of its components:

Letβs keep note of this. Next we will find the horizontal and vertical components of the 60 N force .

Once again, we can draw the horizontal and vertical components of this force onto our diagram.

And we can use the same trigonometric equations to figure out the magnitudes of the horizontal and vertical components. This time, H is 60 and is :

So, we can write as the sum of its components (rounded to 2 decimal places) too as follows:

Now, we have the horizontal and vertical components of both red vectors.

Finding the components of these two vectors means we can easily find the components of the resultant, . We can see that the horizontal component of is the sum of the horizontal components of and . Similarly, the vertical component of is the sum of the vertical components of and :

Rounded to 2 decimal places, we have

Now that we have found the components of the resultant force , we can calculate its magnitude. Recall that the magnitude of any 2-dimensional vector is given by where and are the horizontal and vertical components of respectively.

In this case, we have

Rounding this to the nearest newton gives us a final answer of 118 N.

### Key Points

- A full description of certain physical quantities such as displacement,
velocity, acceleration, and force must include information about both magnitude
*and*direction. This makes**vectors**the perfect mathematical tool to describe them. - Any 2D vector can be expressed as the sum of its horizontal and vertical components:
- Given the components of a vector, its magnitude and direction are given by
- Given the magnitude of a vector and the angle at which it acts, we can determine the magnitudes of the horizontal and vertical components of that vector using trigonometry. In this case,