Vectors and Coordinate Systems

Vectors and Coordinate Systems
Chapter Goal: To learn how vectors are represented and used. Slide 3-2

Characteristics of Scalars
Scalar: is a quanity that has magnitude only. ( A number that represents “how much” of something ther is. Examples include: Currency: $5 Temperature: 25 Kelvin Objects/things: 15 people Mass: 20kg Speed: 5 m/s

A vector is a quantity that has both magnitude and direction
A vector is a quantity that has both magnitude and direction. (“How much” and “which way”). It is represented by an arrow. The length of the vector represents the magnitude and the arrow indicates the direction of the vector. Blue and orange vectors have same magnitude but different direction. Blue and purple vectors have same magnitude and direction so they are equal. Blue and green vectors have same direction but different magnitude. Two vectors are equal if they have the same direction and magnitude (length).

Characteristics of Vectors
A Vector is something that has two and only two defining characteristics: Magnitude: the 'size' or 'quantity' Direction: the vector is directed from one place to another. Examples Include: Displacement: 20 meters west Velocity: – 5 m/s

Speed vs Velocity Speed vs. Velocity
Speed is a scalar, (magnitude no direction) - such as 5 feet per second. Speed does not tell the direction the object is moving. All that we know from the speed is the magnitude of the movement. Velocity, is a vector (both magnitude and direction) – such as 5 ft/s Eastward. It tells you the magnitude of the movement, 5 ft/s, as well as the direction which is Eastward.

YOU CAN TRANSLATE VECTORS BUT YOU CAN NOT ROTATE
Vector Drawing Rules 1. The vector is drawn pointing in the direction of the vector. This is probably the key feature of what makes a vector a vector... direction. If an object is moving East, you better make sure that the arrow points East. Always remember that when a direction is written down with the magnitude of a measurement 2. The length of a vector is proportional to the magnitude of the measurement. This just means that the bigger your measurement, the bigger your vector. If I wanted to show you a vector for a car moving at 10 km/h [East], and another one moving at 20 km/h [East], the second vector would be twice as big. 3. A vector can be picked up and moved around in a vector diagram, as long as when you place it in its new location it is still the same size and pointing in the same direction. This is actually a big deal, since we will see later that moving vectors around to arrange them is critical to solving problems. YOU CAN TRANSLATE VECTORS BUT YOU CAN NOT ROTATE 4. There are very specific rules for how vectors can touch each other, most of which will make more sense by the end of this lesson. Vectors must touch head-to-tail (aka tip-to-tail) when you are adding them. Only resultants can touch head-to-head and tail-to-tail. Like I said, more on this soon.

Adding Vectors In One Dimension
Example 1: Sketch a diagram that shows how you would add the following two vectors, A and B. Right now A and B. are not touching head-to-tail, so we will have to rearrange them. Keep in mind that I can pick up and move around the vectors as long as I make sure that when they are plunked back down they are still pointing the same direction and the same size. or

Adding Vectors In One Dimension
The idea that I get the same result when I add the vectors shown above is a very important one in Physics. ● It would be like if I told you that I was planning to walk 4 km East, and then another 6 km East. It would give me the same result (10 km East) as I would get by first walking 6 km and then 4 km. ● Since the end result is 10 km East, I would say my resultant is 10 km East. ● A resultant is the sum total of two or more vectors added. - It shows you what you would get as an end result of the other vectors put together.

Example 2: Sketch the resultant of the addition of the two vectors in Example 1.
Since we already figured out that A + B = B + A we can draw the resultant for either combination and get the same thing. We'll just go ahead and draw it for A + B. The resultant is just another vector drawn as an arrow. The resultant has to start at the beginning (the tail of A) and finish at the end (the head of B). That's the only way for the resultant to show that it has the same overall result as the two original vectors.

Subtracting Vectors in One Dimension
This is where things get a bit more interesting. ● What we need to remember here is that in Physics a negative sign simply means “in the opposite direction.” ● We can take A - B and simply change it into A + -B. ● The negative sign on the B just means that we will need to take the original vector B and point it in exactly the opposite direction (180° from where it's pointing originally). ● Then we will simply add them just like we did in the previous diagrams (touching head to- tail of course!) to get our resultant. ● The reason you may have to do subtraction of vectors is because some physics formulas require you to subtract vectors. For example, Δv = vf – vi.

Subtracting Vectors in One Dimension
Example 3: 1. Sketch a vector diagram of A – B. The most important thing to remember is that A - B is equal to A + -B. So, all we need to do is take the original vector for B and spin it around so it points in the opposite direction. 2. Now we just add them head-to-tail and draw in our resultant. Be careful since subtraction is not commutative (that just means that A - B ≠ B - A).

Vectors in Two Dimensions
Two dimensional problems are a little tougher, because we are no longer just lining up one dimensional vectors and doing quick math. ● Instead, we need to pay attention to how 2D vectors form a more complex (but not very complex) diagram. This will involve right angle triangles (in this class..no law of sins or cosines). ● If they are right angle triangles, just use your regular trig (SOH CAH TOA) and pythagoras (c2 = a2 + b2). ● You'll want to be thinking about physics as you set up your diagram (so that you get everything pointing head-to-tail and stuff) and then switch over to doing math just like any trig problem.

Components of Vectors One of the most important ideas in vectors is components. ● Just like a component stereo system is made of several individual parts working together. ● Components of vectors are the individual parts that add up to the overall resultant.

Example 1: A car drives 10.0 km [E] and then 7.0 km [N]. Determine its displacement. First, draw a proper diagram... The red and the blue vectors are the components of the resultant: ● The red and blue components show you how walking East and the North will result in you moving more or less in a North-East direction. ● 10 km [E] is shown leading up to 7.0 km [N]. Start at the tail of the red arrow and follow the path it takes you along. You eventually end up at the head of the blue vector. ● The resultant is drawn in head-to-head and tail-to-tail, just like a resultant is always supposed to be. ● This is certainly a right angle triangle, so just use c2 = a2 + b2 to find out the magnitude (size) of the resultant. c2=a2+b2 c2= c= =12 km

2D Vector Direction Since this problem is about vectors we would also have to give a direction in our answer. ● This will involve measuring an angle, but it will also involve a way to make sure everyone knows what that angle means!!!

Head to Tail Method To add vectors, one method is to do “head to tail” addition. Terminal point (head) of w Move w over keeping the magnitude and direction the same. Initial point (tail) of v

2D Vector Direction The problem with angles is that on their own they don't mean very much. If I say to you “20o ”, you need to ask “20o pointing where?” Over the years there have basically been two systems that are accepted for giving meaning to the angles used in Physics: 1. Cartesian Method 2. Navigator Method Cartesian Method (just like math class) It divides space into four quadrants using an x and y axis. The positive x axis is considered to be the beginning and is assigned an angle of 0o. As you move counter-clockwise, the angles get bigger. At each axis point you have added another 90o. The Cartesian method is fine for math class, but it does have one serious drawback; the positive x axis being set as 0o is an arbitrary (for no good reason) decision. As a result, in most cases it will not be used in your book or in my notes as often. However, this does not mean that you can ignore it.

2D Vector Direction Navigator Method The navigator method gets rid of the arbitrary nature of the Cartesian method by using compass points. We will use one of the four compass points as our reference, and then measure how many degrees towards one of the other compass points we have moved. We do have to be a little careful, because when using the navigator method there are two different styles of writing down the answer, and every answer can be given using two different angles! The following examples will show you how this can work.

2D Vector Direction Example 2: Identify the direction of the vector shown in the illustration. If you look at where the angle is placed in this diagram, you'll probably agree that we are measuring an angle that is 300 away from the North. ● It's like we are using North as a reference line in this situation. The vector is 30°away from North. Looking at the diagram, we're basically going clockwise towards East a little bit. ● So, we know that we are using North as a direction, but we're leaning over towards East somewhat. In the physics style of giving a direction, we would write 300 East of North ● However, we could also say we are 60o North of East. ● Which is correct. BOTH ARE. However, your book seems to use E and W as the reference line and N and S as the reference from line.

Example The direction of the vector is 55° North of East (or 350 east of North) The magnitude of the vector is 2.3.

Adding Vectors by Components
Break down the Vector in to its x and y components using trig x y V Vx Vy θ

We move/translate component vectors(notice vy below) all the time in physics to better solve and explain problems. x y V Vx Vy θ

Moving Between the Geometric Representation and the Component Representation
If a component vector points left (or down), you must manually insert a minus sign in front of the component, as done for By in the figure to the right. The role of sines and cosines can be reversed, depending upon which angle is used to define the direction. We Use the Reference Angle 95% of time in physics but it really does not matter Slide 3-40

If a ball is thrown with a velocity of 25 m/s at an angle of 37° above the horizontal, what is the vertical component of the velocity? A 12 m/s B 15 m/s C 20 m/s [This object is a pull tab] Answer B D 25 m/s

If a ball is thrown with a velocity of 25 m/s at an angle of 37° above the horizontal, what is the horizontal component of the velocity? A 12 m/s B 15 m/s C 20 m/s [This object is a pull tab] Answer C D 25 m/s

If you walk 6.0 km in a straight line in a direction north of east and you end up 2.0 km north and several kilometers east. How many degrees north of east have you walked? A 19° B 45° [This object is a pull tab] Answer A C 60° D 71°

Adding 2 Vectors by Components
Using the tip-to-tail method, we can sketch the resultant of any two vectors. x y V1 V2 V But we cannot find the magnitude of ‘V', the resultant,  since V1 and V2 are two-dimensional vectors

Adding 2 Vectors by Components
Add vectors A and B

Adding Vectors On a graph, add vectors using the “head-to-tail” rule:
Move B so that the head of A touches the tail of B Note: “moving” B does not change it. A vector is only defined by its magnitude and direction, not starting location.

Adding Vectors The vector starting at the tail of A and ending at the head of B is C, the sum (or resultant) of A and B.

Adding Vectors The negative of a vector is just a vector going the opposite way. A number multiplied in front of a vector is called a scalar. It means to take the vector and add together that many times.

Using the vectors shown, find the following:

Example 1: On a hot summer day a person goes for a walk to see if they can find a 7-Eleven to buy a Slurpee. He first walks 3.5 km [N], then 4.2 km [E] , and finally 1.4 km [S] before getting to the 7-Eleven. Oh, thank heaven! Determine the displacement of the person. 1. A quick sketch will help us organize things so we can get to work.

Example 1: On a hot summer day a person goes for a walk to see if they can find a 7-Eleven to buy a Slurpee. He first walks 3.5 km [N], then 4.2 km [E] , and finally 1.4 km [S] before getting to the 7-Eleven. Oh, thank heaven! Determine the displacement of the person.

Example 1: On a hot summer day a person goes for a walk to see if they can find a 7-Eleven to buy a Slurpee. He first walks 3.5 km [N], then 4.2 km [E] , and finally 1.4 km [S] before getting to the 7-Eleven. Oh, thank heaven! Determine the displacement of the person. c2=a2+ b2 c2= c= c=4.7km tanθ=opp’adj tanθ=4.2/2.1 θ= O θ=63O The person walked a displacement of 4.7 km 630 N of E.

Example 2: A wagon is being pulled by a rope that makes a 25o angle with the ground. The person is pulling with a force of 103 N along the rope. Determine the horizontal and vertical components of the vector. (i.e. find the x-component and y-component) Draw a quick sketch to help organize your thoughts. Then you can start using trig to find the components.

Example 2: A wagon is being pulled by a rope that makes a 25o angle with the ground. The person is pulling with a force of 103 N along the rope. Determine the horizontal and vertical components of the vector. (i.e. find the x-component and y-component)

Example 2: A wagon is being pulled by a rope that makes a 25o angle with the ground. The person is pulling with a force of 103 N along the rope. Determine the horizontal and vertical components of the vector. (i.e. find the x-component and y-component) Or in x and y coordinates x= 93N y= 44N

Solving 2 Vectors Adding Using Component Vectors!!
We can add vectors together by using the x and y components of each vector. Two Methods Tail to Head Method Parallelogram Method We will be using this method all semester long

Solving Multiple Vectors Using Vector Components
Head to Tail method to Solve Multiple Vector Problems We now know how to break V1 and V2 into components... y V2y V2 V1 V2x V1y x V1x

Solving Multiple Vectors Using Vector Components
And since the x and y components are one dimensional,  they can be added as such. y V2y Vy = V1y + V2y V1y V1x V2x x Vx = V1x + V2x

1.  Draw a diagram and add the   vectors graphically. V2 V1

1.  Draw a diagram and add the    vectors graphically using the head to tail method. 2.  Choose x and y axes. V2 V1 x

1.  Draw a diagram and add the   vectors graphically. 2.  Choose x and y axes. 3. Resolve each vector into x    and y components. V2 V2y V1 V2x V1y V1x x

1.  Draw a diagram and add the   vectors graphically. 2.  Choose x and y axes. 3. Resolve each vector into x   and y components. 4. Calculate each component. V2 V2y V1 V2x V1y V1x x V1x = V1 cos(θ1) V2x = V2 cos(θ2)  V1y = V1 sin(θ1) V2y = V2 sin(θ2)

Vx 1.  Draw a diagram and add the   vectors graphically. 2.  Choose x and y axes. 3. Resolve each vector into x   and y components. 4. Calculate each component. 5. Add the components in each   direction. V2 V2y Vy V1 V2x V1y V1x x

x y V1 V1x V1y V2 V2y V2x V Vx Vy 1.  Draw a diagram and add the   vectors graphically. 2.  Choose x and y axes. 3. Resolve each vector into x   and y components. 4. Calculate each component. 5. Add the components in each   direction. 6.  Find the length and direction of the resultant vector.

Parallelogram Rule for Vector Addition
It is often convenient to draw two vectors with their tails together, as shown in (a) below. To evaluate F  D  E, you could move E over and use the tip-to-tail rule, as shown in (b) below. Alternatively, F  D  E can be found as the diagonal of the parallelogram defined by D and E, as shown in (c) below. Slide 3-24

Parallelogram Method

Example: Graphically determine the resultant of the following three vector  displacements:   m, 30º north of east   m, 37º east of north   m, 50º west of south

Example: Graphically determine the resultant of the following three vector  displacements:   m, 30º north of east   m, 37º east of north   m, 50º west of south 30O 37O 50O 24m 28m 20m First, draw the vectors.

Next, find the x and y components of each.
Example: Graphically determine the resultant of the following three vector  displacements:   m, 30º north of east   m, 37º east of north   m, 50º west of south 30O 37O 50O 24m 28m 20m Next, find the x and y components of each.

Next, find the x and y components of each.
Example: Graphically determine the resultant of the following three vector  displacements:   m, 30º north of east   m, 37º east of north   m, 50º west of south 30O 37O 50O 24m 28m 20m Next, find the x and y components of each. Note: these are both negative because south and west are negative directions.

Now we can put the components in a chart and solve for the resultant vector.
To find the magnitude of the resultant, use the Pythagorean theorem. x (m) y (m) d1 20.8 12.0 d2 16.9 22.4 d3 -15.3 -12.9 Ʃ 21.5 To find the direction of the resultant, use inverse tangent.

Which of the following is an accurate statement? A A vector cannot have a magnitude of zero if one of its components is not zero. B The magnitude of a vector can be equal to less than the magnitude of one of its components. C If the magnitude of vector A is less than the  magnitude of vector B, then the x-component of A must be less than the x-component of B. D The magnitude of a vector can be either  positive or negative. [This object is a pull tab] Answer A

Unit Vectors Each vector in the figure to the right has a magnitude of 1, no units, and is parallel to a coordinate axis. A vector with these properties is called a unit vector. These unit vectors have the special symbols: Unit vectors establish the directions of the positive axes of the coordinate system. Slide 3-45

Vector Algebra When decomposing a vector, unit vectors provide a useful way to write component vectors: The full decomposition of the vector A can then be written: Slide 3-46

Example 3.5 Run Rabbit Run! Slide 3-49

Example 3.5 Run Rabbit Run! Slide 3-50

Vectors and Coordinate Systems

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