Working With Vectors

Size and Direction Matter!

Presentation 2003 R. McDermott

Vectors and Scalars

• Scalars have size (magnitude), but direction doesn’t matter:

• Distance, speed, mass, time

• Vectors have magnitude, but direction does matter:• Displacement, velocity, force, acceleration

• Vectors are concerned with where you end relative to where you start

• Scalars are concerned with how you got there

Vectors

• Let’s exert a force on a box by attaching a rope and pulling:

• What do you think would happen?

Enough Rope• Suppose we apply a second force (attach a

second rope), what do you think will happen?

• Do we all agree?

Direction Matters!

• Well, here’s one possibility:

• And here’s another:

Direction Really Matters!• Here’s the top view of a third possibility!

• There are, in fact, an infinite number of possible sums of two forces!

Vector “Addition”

• Ok, let’s start with the easy ones:• Vectors in the same direction (angle of 0º)• Vectors opposite (angle of 180º)

• The sum on the left is: 5 right + 8 right = 13 right

• The sum on the right is: 5 right + 8 left = 3 left

5 8 58

“Addition” cont.

• Because direction matters, we have to specify direction when adding. We can also do this by using + and – signs:

5 + 8 = 13 5 + (-8) = -3

• These answers provide the maximum and minimum sum of these two vectors

• But what if the vectors are at angles?

Tip-to-Tail

• To add two (or more) vectors, arrange them so they are tip to tail as below:

• The sum is then found by drawing a vector from the tail of the first vector to the tip of the last, and is called the resultant

Finding the Resultant:

• The value of the resultant can be found graphically (as below), or mathematically

• Since this is a right triangle, we can solve using the Pythagorean Theorem

Another Example:

• To add two (or more) vectors, arrange them so they are tip to tail as below:

• The sum is then found by drawing a vector from the tail of the first vector to the tip of the last

Tail-to-Tail

• That was the triangle method. You can also use the parallelogram method

• Complete the parallelogram and draw in the diagonal from your starting point:

More Than Two Vectors:

General Case:

• For two vectors of 5 and 8, you can “add” and get a maximum resultant of 13 to the left or right (angle 0°)

• Or you can “add” to get a minimum resultant of 3 to the left or right (angle 180°)

• Or you can “add” and get any resultant value between 3 and 13 (angle between 0° and 180°)!

• You cannot, however, get 2 or 15!

• What happens to the value of the resultant as the angle increases from 20° to 60°?

Resultant and Angle:

Distance

• Distance is how far you move all together

• The distance traveled in the diagram is:

100m + 60m + 40m + 100m + 60m = 360m

Displacement - Resultant

• Displacement is direct from start to finish

• 40m to the left or –40m

• Distance may not equal displacement!

R

Speed – Distance/time

• If a woman drove this pattern of roads in 20 seconds, her average speed would be:

360m/20s = 18m/s

Velocity – Displacement/Time

• But her average velocity would be:

40m/20s = 2m/s to the left, or –2m/s

• Velocity and speed do not have to be the same!

R

Gimme Some Direction!

Vector Components• The components of a vector are its projections onto

a set of axes (usually vertical and horizontal):

• For the vector V above, what are the values (in symbols) of its projections, VH and VV?

V

Finding Components

• First construct perpendicular lines from the end of V to the axes:

• Next, draw vectors from the origin to the point where the perpendiculars met the axes

V

Components

• This is the horizontal component VH:

• And this is the vertical component VV:

V

VH

VV

Component Values:

• From the diagram, VH = Vcos

• And similarly, VV = Vsin

V

Vcos

Vsin

What Are Components?

• Rearranging the diagram, we can see that the sum of the components is V!!

V = VH + VV = Vcos + Vsin

V

Doggone Vectors!

The Total is the Sum of the Pieces

How Are They Used?

• We can now add vectors by finding their components, adding the vertical parts, adding the horizontal parts, and then using the Pythagorean Theorem to find their resultant (sum).

• In other words, we can change two-dimensional problems with angles into linear problems that are much easier to handle.

• This is a fundamental technique in solving physics problems (as we will see)!