Vector

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 DIFFERENTVECTOR!!!

• 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?

300 [m]

400 [m]

500 [m]

Triangle Method Example 1

300 [m]

400 [m]

500 [m]

The resultant is 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.

Or this: Or this:Like this:

Resultant

To get the resultant, we draw an arrow from the corner with two tails to the corner with two heads.

Or this: Or this:Like 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.

θ

Our original vector is at an angle, θ, with respect to the positive x-axis.

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 vector . It has magnitude |A|.Av

The magnitude of the x-component, Ax, is given by the equation

θcosAAx =

The magnitude of the y-component, Ay, is given by the equation

θsinAAy =

Also, the ratio, is given by the equation

θtan=x

yA

A

xy AA

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 y-components together.

•Once we have a resultant x-component and a resultant y-component, 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 x-components:

and the y-components

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.