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

“Addition” cont. Because direction matters, we have to specify direction when adding. We can also do this by using + and – signs: = (-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, V H and V V ? 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 V H : And this is the vertical component V V : V  VHVH V

Component Values: From the diagram, V H = Vcos  And similarly, V V = Vsin  V  Vcos  Vsin 

What Are Components? Rearranging the diagram, we can see that the sum of the components is V!! V = V H + V V = 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)!