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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)!