In this example we
will be adding the two vectors shown below using the
component method. The vectors we will be adding are
displacement vectors, but the method is the same with any
other type of vectors, such as velocity, acceleration, or
force vectors.

There is an vector component addition calculator below.

Click hereto go to the bottom of this
page where you will find a calculator to help you check your
work. The intermediate calculations in the calculator are not
rounded as much as those presented below, so, the numbers
will differ slightly. The numbers in the following
explanation are rounded so as to not clutter the
presentation.

Here is a diagram showing the method we are going to explain.

These are the vectors that we are going to add.

Some of the
subscript notation used in the graphics is very slightly
different than the notation used in the text. The drawing
program used to generate the graphics does not easily support
subscripts, so, you may see notation such as 'd1' in a
picture and notation such as 'd_{1}' in the text. Hopefully, this
small difference will not cause any problems.

The
component method of addition can be summarized this way:

Using
trigonometry, find the x-component and the
y-component for each vector. Refer to a diagram
of each vector to correctly reason the sign, (+
or -), for each component.

Add up both
x-components, (one from each vector), to get the
x-component of the total.

Add up both
y-components, (one from each vector), to get the
y-component of the total.

Add the
x-component of the total to the y-component of
the total, and then use the Pythagorean theorem and
trigonometry to get the size and direction of the
total.

Right triangle trigonometry is used to find the separate components.

Let's take this all
one step at a time. First, let's visualize the x-component
and the y-component of d_{1}.
Here is that diagram showing the x-component in
red and the y-component in
green:

The two components along with the original
vector form a right triangle. Therefore,
we can use right triangle trigonometry to find the lengths of the two components.
That is, we can use the 'SOH-CAH-TOA' type
of definitions for the sine, cosine, and
tangent trigonometry functions.

Finding the first x-component.

Let's find the
x-component of d_{1}. Notice that the
x-component is adjacent to the angle of 34 degrees, so, we
will use the cosine function since it relates an acute angle,
the adjacent side to that angle, and the hypotenuse of a
right triangle:

Now, using
trigonometry like this will not
tell us the sign, (+ or -), of this component, (or any
other). So, we must check the diagram for positive or
negative directions. This x-component is aimed to the right,
so, it is positive:

(Again, remember
that these calculations presented here have decimals that
have been truncated. Presenting calculations to many more
decimals does not help clarify methods, and, also, it
violates several rules of significant digits. In other words,
these calculations are approximate. The calculator below
keeps many more decimal places, so, its outputs will differ
slightly.)

Finding the first y-component.

Now, let's find the
y-component of d_{1}. Notice that the
y-component is opposite to the angle of 34 degrees, so, we
will use the sine function since it relates an acute angle,
the opposite side to that angle, and the hypotenuse of a
right triangle:

Again, check the
diagram for positive or negative directions. The y-component
aims up, so, it is positive:

The calculated values for the first set of components

Here is the diagram
now showing the values for the x-component and y-component of
d_{1}:

Visualizing the components for the second vector.

Next, let's see the
x-component and the y-component of d_{2}.
Here is that diagram showing the x-component in
red and the y-component in
green:

Calculating the second x-component

We will now find the
x-component of d_{2}. Here the
x-component is opposite the angle of 64 degrees, so we will
use the sine function to find it:

Check the diagram
for positive or negative directions. This x-component points
to the left, so, it is negative:

Calculating the second y-component

And for our last
component we will find the y-component of d_{2}.
The x-component is adjacent to the angle of 64 degrees in
this diagram, so we will use the cosine function to find it:

Check the diagram
for positive or negative directions. This y-component is
aimed up, so, it is positive:

Final values for the second set of components

Here is the diagram
showing our newly calculated values for the components of d_{2}:

Add all the x-components, add all the y-components, but
do not add an x- to a y-component.

Now, we must add up
like components to get the components for the total
displacement.

To get the total
x-displacement, add up all of the separate x-components:

To get the total
y-displacement, add up all of the separate y-components:

So, when these two
vectors, d_{1} and d_{2},
are added, the total, or sum, has an x-component of 9.2
meters and a y-component of 30.1 meters.

Put together the components of the total to create the
total two dimensional vector.

To get the actual
2-D total displacement, add the total x-displacement and the
total y-displacement. Here is a diagram with the total
x-component shown in red and the total y-component
shown in green and the 2-D total shown in
blue:

Use
the Pythagorean theorem to get the magnitude (size) of the
total 2-D displacement:

Use the arctangent
function to get the angle:

Check the diagram
for NSEW notation:

Final result

Therefore, our final
result for the total 2-D displacement can be stated as:

Here's a diagram that shows this result:

Summary

Again, the component
method of addition can be summarized this way:

Using
trigonometry, find the x-component and the
y-component for each vector. Refer to a diagram
of each vector to correctly reason the sign, (+
or -), for each component.

Add up both
x-components, (one from each vector), to get the
x-component of the total.

Add up both
y-components, (one from each vector), to get the
y-component of the total.

Add the
x-component of the total to the y-component of
the total then use the Pythagorean theorem and
trigonometry to get the size and direction of the
total.

Vector
Adder, Component Method

Make up your own vector addition problem.
Create two vectors and write down their magnitudes and
directions. Then, using the above described methods find
their sum. Check your answer using the calculator below.

To use this calculator enter the magnitude
and direction of the first and second vectors. Be sure to
correctly click the radio buttons that tell the orientation
for each of the direction angles. Click the 'Calculate Sum'
button. The x- and y-components for each vector, the total x-
and y-component for the sum, and the final magnitude and
direction of the sum will be presented in a pop-up message
box.

Click this button to see the computer code for this calculator.

Here's the code that runs the above calculator.

You can change the code, if you like, and then click the following
'Reevaluate code' button.
The program will then work as per your changes.
Of course, your changes, especially random changes, can introduce
errors, miscalculations, and browser crashes. If you
need to get things back to their original condition, just reload
this page using your browser's reload button.

The intention here is to conveniently show the inner workings
of this program so that you understand how the calculation is done.

Click the 'Code' button again to close this section.