Adding Vectors in Two
Dimensions
Triangle Method
Heads or Tails?
If we want to add two vectors, such as those below,
we must first make the head of one vector touch the
tail of the other.
Like this:
Or this:
Can Only Slide
• Notice how when I made the head of one
vector touch the tail of the other, I did not
change their lengths or their directions.
– CHANGING THE MAGNITUDE OR THE
DIRECTION MAKES A DIFFERENT
VECTOR!!!
• We must “slide” vectors, preserving their
magnitudes and directions.
Finding the Resultant
Once you’ve “slid” your vectors into place (head to
tail), making sure that you didn’t change the
magnitude or direction of either one, you can find the
resultant by drawing an arrow from the free tail to the
free head.
Like this:
Or this:
Notice that in both cases, we get the same resultant.
Well, that makes sense considering we used the same
two vectors.
Putting It All Together
• To see the entire triangle method of vector
addition in action, check out this
simulation.
• Notice that the triangle method will work
for any number of vectors. (Although with
more than two, you no longer form a
triangle.
Triangle Method Example 1
• A person travels 300 [m] North and 400
[m] East. What is the magnitude of their
displacement?
500 [m]
300 [m]
400 [m]
Triangle Method Example 1
500 [m]
?
400 [m]
The
resultant is
300 [m] how many
degrees
North of
East?
Triangle Method Example 2
• A ship travels 200 [km] at a heading 25°
NE then 400 [km] at a heading 55° NW.
What is the magnitude and displacement of
the ship?
HINT: Use the Law of
Cosines
Adding Vectors in Two
Dimensions
Parallelogram Method
Orientation
Unlike the triangle method of vector addition, the
parallelogram method of vector addition doesn’t care what
orientation the vectors have.
Vectors can be head to tail, head to head, or
tail to tail.
OR
OR
Complete the Parallelogram
Now, you make “copies” of the original vectors, so that
you complete a parallelogram.
Like this:
Or this:
Or this:
Resultant
To get the resultant, we draw an arrow from the corner
with two tails to the corner with two heads.
Like this:
Or this:
Or this:
Notice that no matter what our original orientation was,
we got the same resultant.
Need another look? Check out this simulation.
Parallelogram Method Example
• Two forces are acting concurrently (on the
same point at the same time) on a box. F1 =
40 N East F2 = 30 N South. What is the
resultant force on the box?
F1
F2
F1 + F2 = 50 [N], 37° SE
Making the Connection
Check out what happens when we overlap the two
resultants from the triangle method.
It’s the parallelogram method all over again!
Adding Vectors in Two
Dimensions
Analytic Method
Resolving Vectors
• Vectors can be “resolved” into their
components.
• In other words, we can make a vector
(that’s not along one of the axes) the
hypotenuse of a right triangle.
• Using trigonometry, we can then figure out
what each component is.
For Example
Now, we have one component in the x-direction and
one component in the y-direction, making the angle
between them 90°.
Notice that our original
vector is the resultant of it’s
components.
Trig at Work
We know the directions of our components, but how do
we find their magnitudes? That’s where trigonometry
comes into play.
Our original vector is at an
angle, θ, with respect to the
positive x-axis.
θ
We can find the x and y components of the vector using
trigonometric functions (sine, cosine, and tangent).
Trig at Work
Let’s call our original
v
vector A .
It has magnitude |A|.
The magnitude of the
x-component, Ax, is
given by the equation
The magnitude of the
y-component, Ay, is
given by the equation
Ax = A cosθ
Ay = A sin θ
Also, the ratio, Ay
Ax
is given by the equation
Ay
Ax
= tan θ
Resolution Example
• Resolve a vector of magnitude 100 [km] at
a direction of 60° NE into its x and y
components.
d1 = 69 km North
d2 = 50 km East
How Does This Help Us Add
Vectors?
•Remember how easy it was to add vectors when they
were only in one dimension? All you had to do was add
the magnitudes, making sure you used the right sign for
direction.
•Now, you can do that for each dimension separately. In
other words, add the x-components together and add the ycomponents together.
•Once we have a resultant x-component and a resultant ycomponent, we can just use the Pythagorean theorem to
find the magnitude of the resultant and one of our
trigonometric equations to find the angle.
For Example
We want to add the two vectors below.
We add the xcomponents:
and the ycomponents
Example
Now, we put our resultant components together to get the
total resultant vector.
We can get it’s magnitude using the Pythagorean
theorem and it’s angle using trigonometry.
Analytic Method Example
• A ship travels 200 [km] at a heading 25°
NE then 400 [km] at a heading 55° NW.
What is the magnitude and displacement of
the ship?
• Compare this answer to the one you
obtained when solving it via the triangle
method.