Vectors This is one of the most important chapters in the course. PowerPoint presentations are compiled from Walker 3 rd Edition Instructor CD-ROM and Dr. Daniel Bullock’s own resources Learning objectives Scalars Versus Vectors The Components of a Vector Adding and Subtracting Vectors Unit Vectors Position, Displacement, Velocity, and Acceleration Vectors Relative Motion

Scalars and Vectors Scalar  has only magnitude (size) –5 seconds (time) –10 kg (mass) –13.6 eV (energy) –10 m/s (speed) Vector  has both magnitude (size) and direction –10 m/s North (velocity) –- 5 m/s 2 (acceleration) –4.5 Newtons 37 0 North of West (force)

Vectors Vector has tail (beginning) and tip (end) A position vector describes the location of a point Can resolve vector into perpendicular components using a two-dimensional coordinate system: tail tip

Vectors Length, angle, and components can be calculated from each other using trigonometry: Length (magnitude) of the vector A is given by the Pythagorean theorem

Vectors Signs of vector components

Adding and Subtracting Vectors Adding vectors graphically: Place the tail of the second at the head of the first. The sum points from the tail of the first to the head of the last. Adding Vectors Using Components: 1. Find the components of each vector to be added. 2. Add the x- and y-components separately. 3. Find the resultant vector.

Adding and Subtracting Vectors Subtracting Vectors: The negative of a vector is a vector of the same magnitude pointing in the opposite direction. Here, D = A – B.

Unit Vectors Unit vectors are dimensionless vectors of unit length. Multiplying unit vectors by scalars: the multiplier changes the length, and the sign indicates the direction.

Vector Addition A motor boat is moving 15 km/hr relative to the water. The river current is 10 km/hr downstream. How fast does the boat go (relative to the shore) upstream and downstream? Boat Upstream Vector

Vector Addition A motor boat is moving 15 km/hr relative to the water. The river current is 10 km/hr downstream. How fast does the boat go (relative to the shore) upstream and downstream? Boat Upstream Vector Boat Downstream Vector

Vector Addition A motor boat is moving 15 km/hr relative to the water. The river current is 10 km/hr downstream. How fast does the boat go (relative to the shore) upstream and downstream? Current Vector = 10 km/hr downstream Boat Upstream Vector Boat Downstream Vector

Boat Velocity Upstream Upstream: Place vectors head to tail, net result, 5 km/hr upstream

Boat Velocity Upstream Upstream: Place vectors head to tail,

Boat Velocity Upstream: Place vectors head to tail, net result, 5 km/hr upstream Start Finish Difference

Boat Velocity Downstream: Place vectors head to tail,

Boat Velocity Downstream: Place vectors head to tail, net result,

Boat Velocity Downstream: Place vectors head to tail, net result, 25 km/hr downstream Commutative law

Forces On An Airplane When will it fly? Gravity Propulsion Net Force?

Forces On An Airplane When will it fly? Gravity Propulsion Net Force Plane Dives to the Ground

Forces On An Airplane When will it fly? Gravity Propulsion Lift Net Force?

Friction When will it fly? Gravity Propulsion Lift Net Force = 0 up or down Plane rolls along the runway like a car because of propulsion.

Forces On An Airplane When will it fly? Gravity Propulsion Lift Net Force Plane Flies as long as Lift > Gravity

Friction When will it fly? Gravity Propulsion Lift Air Resistance Net Force = 0 Equilibrium

Flight When will it fly? Gravity Propulsion Lift Air Resistance Net Force Plane Flies as long as Lift > Gravity AND Propulsion > Air Resistance

Adding (and subtracting) vectors by components Let’s say I have two vectors: I want to calculate the vector sum of these vectors: Let’s say the vectors have the following values:

A B A B Our result is consistent with the graphical method! What’s the magnitude of our new vector?

A B + How would you find the angle, , the vector makes with the y-axis?  opp = 2  adj = 12

Multiplying vectors by scalars: So if the vector A was: the scalar, a = 5 then the new vector: and it was multiplied by Scalar Product: (aka dot product): mag. of a mag. of b angle between the vectors

Scalar Product: (aka dot product): vectors scalars The dot product is the product of two quantities: (1)mag. of one vector (2)scalar component of the second vector along the direction of the first

Vector Product (aka cross product) The vector product produces a new vector who’s magnitude is given by: The direction of the new vector is given by the, “right hand rule” Mathematically, we can find the direction using matrix operations. The cross product is determined from three determinants

The determinants are used to find the components of the vector 1 st : Strike out the first column and first row! 3 rd : Strike out the 2 nd column and first row 4 th : Cross multiply the four components,subtract, and multiply by -1: 2 nd : Cross multiply the four components – and subtract: x - component y - component

5 th : Cross out the last column and first row 6 th : Cross multiply and subtract four elements z-component So then the new vector will be: We’ll look more at the scalar product when we talk about angular momentum. Example:

Notice the resultant vector is in the z – direction!

Position, Displacement, Velocity, and Acceleration Vectors Average velocity vector: So v av is in the same direction as Δr.

Position, Displacement, Velocity, and Acceleration Vector Average acceleration vector is in the direction of the change in velocity:

Instantaneous velocity vector is tangent to the path: Position, Displacement, Velocity, and Acceleration Vector

Velocity vector is always in the direction of motion; acceleration vector can point anywhere: Position, Displacement, Velocity, and Acceleration Vector

Relative Motion The speed of the passenger with respect to the ground depends on the relative directions of the passenger’s and train’s speeds:

Relative Motion This also works in two dimensions:

Chapter 3 Summary Scalar: number, with appropriate units Vector: quantity with magnitude and direction Vector components: A x = A cos θ, B y = B sin θ Magnitude: A = (A x 2 + A y 2 ) 1/2 Direction: θ = tan -1 (A y / A x ) Graphical vector addition: Place tail of second at head of first; sum points from tail of first to head of last

Chapter 3 Summary Component method: add components of individual vectors, then find magnitude and direction Unit vectors are dimensionless and of unit length Position vector points from origin to location Displacement vector points from original position to final position Velocity vector points in direction of motion Acceleration vector points in direction of change of motion Relative motion: v 13 = v 12 + v 23