Vectors

Vector Quantities:

Quantities that require both magnitude and direction to describe them Ex) Displacement, Velocity, Acceleration, Force

Scalar Quantities:

Quantities that require only magnitude to describe them

Ex) Time, mass, temperature, distance, speed,

Representing Vector Quantities

Vector quantities are often represented by scaled vector diagrams. Vector diagrams depict a vector by use of an arrow

drawn to scale in a specific direction.

The magnitude of the vector is represented by the length of the line and the direction is represented by the arrow “head”

Adding Vectors

Resultant:

The sum of two or more vector quantities

Adding Vectors that are pointing in the same direction

“moving floor” The angle between the vectors is 0o

The resultant equals the sum of the two quantities

It is the maximum resultant you could get between two vector quantities

Adding Vectors that are pointing in the opposite direction

“moving floor” The angle between the vectors is 180o

The resultant equals the difference between the two quantities

It is the minimum resultant you could get between two vector quantities

Example Problem #1

Victoria is swimming in the Erie canal. ( I don’t know why) She can swim at 2.5 m/s however the water in the canal is moving at 1.5 m/s.

What is Victoria’s resultant velocity if she swims with the current?

What is her resultant velocity if she swims against the current?

Example Problem #2

Stefan is pulling on dresser with a force of 80 N towards the east. His brother is pulling on the same dresser with a force of 45 N towards the west. What is the magnitude and direction of the resultant force acting on the dresser?

Adding Vectors that are at angles other than 0o and 180o

Three Methods of adding vectors

1) Graphical Method

2) Pythagorean Method

3) Component Method

Graphical Method

• Using vector diagrams to add vector quantities

Displacement vectors and the graphical method

Velocity Vectors and the Graphical Method

Force Vectors and the Graphical Method

Steps to Adding Vectors Graphically

1) Draw first vector using appropriate scale and pointing in correct direction ( Scale must be included with diagram)

2) Draw second vector, beginning at the end of the first vector (tip to tail) using an appropriate scale and pointing in the correct direction

3) Draw the resultant vector arrow from the beginning of the diagram to the end (draw arrowhead on resultant at end)

4) Measure the length of the resultant and convert using the scale

5) Measure the angle the resultant makes with the first vector arrow.

Example Problem #1

Mark was wandering around the village of Fairport one afternoon. He first walked 5 miles due East then he walked 3 miles at 35o North of East. (a) What was Mark’s resultant displacement? (b) What is the distance Mark traveled?

Example Problem #2

Jake is trying to move a heavy crate across the Wegman’s warehouse floor. He exerts 35 N of force on a rope attached to the crate. His buddy Nick exerts 50 N of force on another rope attached to the crate. If the angle between the concurrent forces is 25o, determine the resultant force exerted by the guys.

Important conclusions about vector addition as learned from the homework

1) The maximum resultant occurs when the angle between the vectors is ___

2) The maximum resultant possible is simply the_________of the vectors

3) The minumim resultant occurs when the angle between the vectors is____

4) The minimum resultant possible is simply the ______________between the vectors

5) As the angle between the vectors increases from 0o to 180o the magnitude of the resultant ____________________

Pythagorean Theorem Method

Using the Pythagorean Theorem and Trig to add vectors

Can be applied only when adding vector quantities that are directed perpendicular to each other

Examples:1. Person swimming across a river

2. A plane flying in a cross wind

Example Problem #1

Bobby is trying to swim across a river that is flowing at 4 m/s. If Bobby can swim at 2.5 m/s, and he makes no correction for the current, what is the magnitude of his resultant velocity?

Example Problem #2

A small airplane is flying at 30 m/s due east in a cross wind of 10 m/s due south. What is the resultant velocity of the airplane?

The Component Method

Vector Components:

A component is a part

These are the projections of the vector along the x- and y-axes

Determining the Components of a Vector

The y-component is moved to the end of the x-component to complete the triangle

Determining the Components of a Vector

The x-component (horizontal component) of a vector is the projection along the x-axis

The y-component (vertical component) of a vector is the projection along the y-axis

cosAxA

sinAyA

Example:

1) Find the components of the velocity of a helicopter traveling 95 km/h at an angle of 35o to the ground.

Recombining Components to Attain the Original Vector

22yx AAA

The magnitude of A is equal to the hypotenuse of the triangle created by the components

The direction can be determined using trig

Ax

Ay

Using the Component Method

1) Break each vector quantity into its x and y components

2) Add x components to x components and y components to y components (take into consideration the directions of the components)

3) Recombine the final x and y components to determine the resultant

Example Problem #1

1) A force of 25 N due east acts concurrently on an object with a force of 35 N acting at 40o N of E. What is the magnitude and direction of the resultant force?