Scalar (Dot) Product

• Many physical relationships can be expressed by the product of vectors, but we cannot use ordinary multiplication with vectors.

• Multiplying vectors using the dot product results in a scalar quantity.

• The dot, or scalar product can be calculated using the magnitudes of the vectors and the angle between them.

• From this definition, it is clear that the dot product is at a maximum when and are parallel () and that the dot product is zero when and are perpendicular.

Scalar Product by Components

• To find the dot product by components, we look at the dot products of the unit vectors.

• Since the dot product of perpendicular vectors is zero,

• But, the dot product of parallel vectors is the product of their magnitudes,

• Multiplying this all out,

• We readily see that the dot product is commutative.

Meaning of Scalar (Dot) Product

• The dot product, , represents the product of the magnitude of and the projection of along the direction of .

• Likewise, it is also the product of the magnitude of and the projection of along the direction of .

Vector (Cross) Product

• Multiplying vectors using the cross product results in a vector.

• The resulting vector is perpendicular to both vectors and .

• The direction is given by the right hand rule. Point the index finger of your right hand along and curl your fingers towards . Your thumb is pointing in the direction of .

• The magnitude of the cross product is.

• The cross product is zero when and are parallel.

Vector Product by Components

• To find the cross product by components, we look at the cross products of the unit vectors.

• Since the cross product of parallel vectors is zero,

• But, using the right hand rule,

• Multiplying this all out, and collecting terms:

• We readily see that the cross product is not commutative.

Vector Product by Determinant

• We can also express the cross product in determinant form.

• Using cofactor (Laplace) expansion, we find the determinant.

Meaning of Vector (Cross) Product

• The magnitude of the cross product, , is the area of a parallelogram with sides and .

• From this, we clearly see that the area (and therefore the cross product) is zero when . and are parallel.

Products of Vectors Example

Chapter 1 SummaryUnits, Physical Quantities, and Vectors

• Idealized models – Know your assumptions

• Units

• SI units, prefixes, and unit consistency

• Uncertainty and significant figures

• Order of magnitude approximations – Is the answer reasonable?

• Vectors (magnitude and direction) and scalars (magnitude)

• Component notation and unit vectors

• Vector addition and subtraction (graphically and by components)

• Dot product:

• Cross product:

Chapter 2 OutlineMotion Along a Straight Line

• Velocity and Acceleration

• Average

• Instantaneous

• Graphical representation

• Motion with constant acceleration

• Kinematic equations

• Free fall

• Motion with varying acceleration

• Equations for position and velocity

Displacement in One Dimension

• First, we need to define a coordinate system.

• For one dimension, this just means choosing the origin and the positive direction.

• Displacement:

• If the displacement is in the positive direction, .

• If the displacement is in the negative direction, .

Average Velocity in One Dimension

• The average velocity is the change in displacement divided by the time interval.

• Is this the same as the average speed?

• Keep in mind that with this definition, only the total displacement and the total time are taken into consideration.

Instantaneous Velocity in One Dimension

• The instantaneous velocity is the limit of the average velocity as the time interval approaches zero.

• Graphically, this is the slope of the position curve.

Finding Velocity on an - Graph

• At any point, the slope of the position vs. time graph will give the instantaneous velocity.

Acceleration

• Just as velocity is the change in position with time, acceleration is the change in velocity with time.

• Similarly, acceleration is the slope of the velocity vs. time graph.

Acceleration vs. Position

• Since acceleration is the time derivative of velocity, and velocity is the time derivative of position, we say that acceleration is the second derivative of position with respect to time.

Acceleration from an - Graph

• Looking at the - graph from earlier, we can glean information about both the velocity and the acceleration.

• Velocity is equal to the slope.

• Upward curvature (concavity):

• Downward curvature (concavity):

• Inflection point:

Motion with Constant Acceleration

• Kinematic equations for constant acceleration.

• Combining to remove time from the equations,

• Combining to remove acceleration from the equations,

• All of these come from the statement that .

Free Falling Bodies

• What happens if we drop an object?

• It will accelerate towards the ground due to the gravitational attraction to the earth.

• Does the acceleration due to gravity depend on the mass or size of the object?

• If the effects of air resistance are small and can be neglected, the acceleration is the same for all bodies.

• We are also disregarding the small corrections due to the earth’s rotation and the distance from the center of the earth.• The acceleration due to gravity, .

• Note that is a positive quantity.

Free Fall Example

Velocity and Position by Integration

• As we saw while deriving the kinematic equations for constant acceleration, we can express velocity as the integral of acceleration and position as the integral of velocity.

• In this case, we will not assume that is a constant.

• Likewise for position and velocity,

Chapter 2 OutlineMotion Along a Straight Line

• Velocity

• Average:

• Instantaneous:

• Slope of position vs. time

• Acceleration

• Average:

• Instantaneous:

• Slope of velocity vs. time; Curvature of position vs. time

Chapter 2 OutlineMotion Along a Straight Line

• Kinematic equations for motion with constant acceleration

• Free fall acceleration due to gravity:

• Motion with varying acceleration