Vectors A vector quantity has both magnitude (size) and direction A scalar quantity only has size (i.e. temperature, time, energy, etc.) Parts of a vector:

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Presentation transcript:

Vectors A vector quantity has both magnitude (size) and direction A scalar quantity only has size (i.e. temperature, time, energy, etc.) Parts of a vector: headtail length – represents the magnitude We can perform math operations with vectors!

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

Vector Components A component of a vector is the projection of the vector on an axis x y Magnitude, size is: We can write the vector A as the sum of an x-component and y-component: A x, A y = the x and y components of the vector A x hat and and y hat are the unit vectors

x y  If we only know the mag. of A, and the angle,  it makes with the x-axis, how do we find the x, and y components?

x y  If we only know the x and y components, how can we find the magnitude of A? This comes from Pythagorean’s theorem GO TO HITT

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 GO TO HITT

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 Go To HITT

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!