1. Or What’s Our Vector Victor?
The Vector Lecture
Or What’s Our Vector Victor?
1b Students know that when forces are balanced, no acceleration occurs, thus an object continues to move at a constant speed or stays at rest.
1i* Students know how to solve 2 dimensional projectile problems
1j* Students know how to resolve two dimensional vectors into their components and calculate the magnitude and direction of a vector from its components.

2. Vectors vs. Scalar Quantities
Vector quantities have both a magnitude (measure) and direction.
Vector quantities add graphically
Vector quantities have components usually perpendicular to one another.
Scalar quantities have direction only
Scalar quantities add algebraically (simply add values)
No components
Can have positive and negative values.
THE TEST

3. Examples Displacement, velocity and acceleration Force
Momentum and Impulse
Torque, angular acceleration, angular velocity and angular displacement
Time, mass
Energy and Work
Rotational kinetic energy

4. Vector Addition
For vectors in same or opposite direction simply add or subtract.
For vectors at right angles use pythagorus to add and get resultant.
For vectors at angles other than right angle break up vector(s) into components, add and then use pythagorus.
Graphically can always use tip to tail and using scaling for magnitude.

5. Vector Decomposition Dividing one vector into two component vectors
Components are amount of vector quantity in that direction
Perpendicular components do not affect one another
Parallel Component = R cosine angle
Perpendicular Component = R sine angle
Opposite of vector addition
R sine angle R
Angle
Question: What does it mean to decompose a vector? What does each component vector represent? What is so special about perpendicular component vectors? What are the formulas used to find perpendicular component vectors?
Activities:
Remind students that vector quantities have some part of them going in one direction and another part of them going in another direction. If we take one vector and make it into two vectors in those directions we have decomposed the vector. The two smaller vectors are called component vectors.
Draw a displacement vector to the northeast and ask how far the hiker traveled to the east and to the north. Draw a coordinate plane and remind students of the sine and cosine functions they’ve used to find the two sides of a triangle given its hypotenuse. We can use the same equations to find the east and north components of this displacement. If necessary, draw a right triangle and derive the sine and cosine ratios. Another way to look at components is that they are the shadow of the resultant vector on the axes.
If useful repeat the above with the weight of a mass on an inclined plane.
4) Draw a vector and show that there are many combinations of components that will give the resultant. State that we often choose to draw perpendicular components because they are independent of one another meaning changing one will not affect the other. We’ll demonstrate this property for motion vectors tomorrow.
5) Point out that decomposition of a vector is the opposite process to the addition of vectors. Here we are trying to find two vectors that when added will give the same effect to a body as the one resultant vector.
R cosine angle

6. Use of Velocity Vectors: Wind and Wave Power
Fastest point of sail is nearly perpendicular to wind
Running with wind means you can only go as fast as the wind
Running sideways to the wind means that wind continues to push on you even when you are going the same speed as wind A B
Question: In what direction is the fastest point of sail for a sail boat?
Activities:
Explain question and then show video (5 minutes)
Explain that moving sideways to the wind means that there is still a force on the sail even when the boat is moving the same speed or faster than the wind.
Same is true for surfing or Boogie Boarding a wave. Go across the wave for fastest speeds. Relate Boogie board story if have time.
Ask question which of the above boats will move the fastest? C

7. Vector vs. Scalar Quantities
Scalars
Vectors
Have magnitude only
Add algebraically
Examples
Time
Mass
Energy
Temperature
Heat and Internal Energy
Represented with normal face type w/o line above value
Magnitude and direction
Add geometrically
Examples
Displacement
Velocity
Acceleration
Force
Momentum
Torque
Represented using arrows where length is magnitude and direction is direction
Represented in text with boldface or with line above value
Question: Define the terms vector an scalar? Which adds geometrically? Give three examples of a vector quantities and three of a scalar quantities used in physics.
Activities:
Define scalar as quantity with only a magnitude. One that can be described fully by a number. Define vector as a quantity with both a magnitude and direction. This means that a vector quantity can be partially in one direction and partially in another like the weight of a mass on an inclined plane or the displacement of a walker.
State that because of direction vectors add differently than scalars. Specifically, 3+4 = 7 in scalar math but 3+4 can equal anything from -1 to 7 in vector math.
Discuss test where place words left, right, down up, north, south with a quantity to see if it makes sense to determine if quantity is a vector or scalar. Try 30 m distance and 2 O Clock North as an example.
List physics scalar quantities
List physics vector quantities
Discuss arrow and written representation of vector quantities. Draw three vectors on board (two of same length but opposite direction and one of the same direction but smaller length), label them a,b, and c and ask which are equal in magnitude and which are in the same direction?
For physics E discuss the meaning of a negative vector (same magnitude but in opposite direction) and what happens to a vector when it is multiplied by a scalar.

8. Check Question 2
Which of the following represents the resulting vector for the two vectors to the right? C B A

9. Check Question 1
Which of the pictured vectors are in the same direction?
A and B
B and C
A and C
Which of the pictured vectors are equal in magnitude? A B C

10. Vector Addition
Vector addition is when two vectors are added to become one resultant vector
To add vectors they must be of the same measurement type and unit.
Graphically add vectors by using tip to tail or parallelogram method
Mathematically adding vectors
Simply add magnitudes if in same direction
Subtract magnitudes if in opposite direction
Use Pythagorean Theorem if at right angles
Direction found using tan q = y/x where y and x are vertical and horizontal measurement of same type
Question: Define Vector addition. How do you find the magnitude and direction of two added vector quantities when they are a) in the same direction b) opposite direction c) at right angles to one another
Activities:
Airplane with tail wind
Airplane with headwind
Airplane w/ crosswind
Airplane w/ combo wind pictorially and describe tip to tail method. Show that could add vectors in any order and come to same place. Show why this is sometimes called the parallelogram rule.
Point out that to add vectors they must be of the same type and unit. In other words add displacements to displacements but not displacements to accelerations and meters can be added to meters but meters cannot be added to kilometers.
For physics E discuss how the direction of the resultant vector can be found using the tangent function and the vertical and horizontal magnitudes. State that can actually use any trigonometric function to find angle depending on what sides of right triangle you have.

11. Check Question 3
Show the perpendicular components of the following resultant vectors. A B D C

12. Vector Addition Revisited
To add vectors not in same direction or at right angles.
Decompose each vector into perpendicular components
Add components in same direction
Find resultant using Pythagorean theorem
Find angle using tangent or other trig function and components
This is where the law of sines and cosines came from Px Py Qx P Q
Question: How can components be used to mathematically add vector quantities which are not parallel or at right angles? Write the final formula for doing this calculation of magnitude. How is the angle of the resulting vector found?
Activities:
Remind students of airplane velocity problem we could not solve mathematically
Draw two vectors, then draw components.
Draw resulting vector and show that sum of vertical components make up adjacent side of right triangle and sum of horizontal components make up opposite side of right triangle
Write distance equation with opposite and adjacent sides, then substitute components and finally take square root of both side. Qy

13. Use of Force Vectors: Inclined Plane
Part of the weight tries to push mass into plane while part of weight tries to pull mass down plane
As the angle of the incline plane goes from horizontal to vertical
The amount of the weight into the plane decreases
The amount of the weight down the plane increases
The overall weight stays the same
Question: What two directions do a masses weight try to push the mass when on an inclined plane? How do each of these change as the incline of the plane increases? Does the actual weight change? Explain why a ball rolls faster down a steep inclined plane vs. a shallow one.
Activities:
Show ball on inclined plane and discuss how weight pushes ball into plane and pulls ball down plane.
Ask what happens to these components as incline angle becomes steeper
Have students make prediction as to whether ball will accelerate more or less when on a steeper incline if their explanation is right. Test it and move on.

14. A Use of Force Vectors: Mechanical Equilibrium
Forces in horizontal direction add to equal zero
Forces in vertical direction add to equal zero
Motion is that of constant velocity (model 1)
one possibility is the object remains stationary
NO ACCELERATION
Examples
String breaks when ends of string are pulled apart
Unsupported chain cannot be pulled straight
Bridge Demo Variables
Question: Under what conditions does mechanical equilibrium occur? What types of motion are possible?
Activities:
Bring out wagon or constant velocity vehicle and lists conditions of equilibrium.
Describe possible motions for wagon, point out that speeding up, slowing down or changing direction are not possibilities.
Show demo with thread holding string and then pull ends apart to show string breaking. Discuss how less of the total force applied is in the vertical direction and more is in the horizontal direction as ends are pulled apart. Point out that to satisfy equilibrium the same force must continue to be applied in the vertical direction. This results in a greater overall force needing to be provided by the string. Use whiteboard if needed.
Do demo with heavy chain showing that one it can never be pulled perfectly straight and two that a small sideways force easily defeats the large forces in line with the chain. Point out similarities with above demo in that the more curved the chain the more force is directed vertically.
Remind students of bridge station yesterday. Ask if increasing bridge height will put more or less of the force in the vertical direction? How does this explain yesterday’s result? How does increasing the distance between the towers affect the amount of force directed vertically?
Pulling car demo is a possibility here or during HW Review

15. Representing Vector Quantities
Text book uses Bold type
Can also use arrow over value
Arrow representation
Length gives magnitude
Direction gives direction

16. Vector Quantities
Vector quantities are those having magnitude and direction. Non-vector quantities are scalar.
A vector quantity partially goes in one direction and partially in another
The test: does it make sense if you put directions after it?
Examples: velocity, acceleration, displacement, force, momentum and torque
Vector quantities add differently than scalar quantities
Question: How does a vector quantity differ from a scalar quantity? Do vector quantities add the same way as scalar quantities? Are the following quantities vectors or scalar: a) time b) mass c) temperature d) displacement e) velocity f) acceleration