Key Terms:

Scalar Quantity: any quantity that has only magnitude (size)

Ex: 35 lbs, 6 ft. long

Vector Quantity: any quantity that has both magnitude (size) & direction

Ex: 30 mph SW, displacement of 450 miles due north

Part 1: Identify the following as either a scalar or vector quantity. Circle one.

  1. My car has traveled 6,000 miles this year.scalar or vector
  2. A plane flew 150 miles due east.scalar or vector
  3. A stone fell down one meter.scalar or vector
  4. She runs 20 miles per week.scalar or vector
  5. My Prius averages nearly 55 miles per gallon of gas.scalar or vector
  6. MACH 1 is the speed of sound at sea level.scalar or vector

Vector quantities are represented by an arrow, referred to as a vector. The length of the vector is proportional to the magnitude of the quantity and the direction gives the direction of the vector quantity.

The symbol (read “vector AB”) represents the vector extending from point A, the initial point, to point B, the terminal point. (Boldface letters and are also used to denote vectors)

The initial point is always said first to indicate direction.

Equivalent Vectors – two vectors that have the same magnitude and
direction

Scalar Multiplication of Vectors – a scalar multiple of is denoted by ,
where is a real scalar.

If , vector points in the opposite direction of vector

Ex: Given , draw and label and .

Resultant Vector – the result of adding two or more vectors together

Head to Tail Method – place the tail of the 2nd vector at the head of the 1st vector and drawn an arrow from the tail of the 1st vector to the head of the 2nd vector

Parallelogram Method –beginning from a common initial point, the 1st vector is added to the 2nd. From the same initial point, the 2nd vector is added to the first. The results vector is drawn to the opposite corner of the parallelogram (if applicable)

THINK: In each grid below, what would it look like if we wanted to find ?

  1. Is vector addition commutative? Is vector subtraction commutative? Provide an example to support your reasoning.

The length of magnitude of a vector is called its norm, and is denoted by .

A vector with a magnitude of 1 is called a unit vector.

Part 2: Find the magnitude of each vector below.

  1. 2.

Component Form of Vectors

Position Vector – a vector draw from the origin to an arbitrary point

Component Form – a position vector written in the form .

Geometry: a vector is a ray

Algebra: a vector is an ordered pair where…

corresponds to the -compontent of the vector

corresponds to the -compontent of the vector

Direction Angle – the positive angle between the -axis and the position vector

Part 3: Draw the resultant vector, find its magnitude, and write each resultant vector below in component form.

  1. 2. 3.

Part 4: Find and the direction angle of vector .

  1. 2. 3.

THINK: Given a vector with a magnitude of 20 and a direction angle of , how would we calculate the horizontal and vertical components of . Write in component form.

Unit Vectors

As detailed earlier, a unit vector is a vector of magnitude 1. Unit Vectors are denoted:

Essentially, we take a vector and shrink the and components, so that the vector is of magnitude 1.

Vectors can also be written in the form , where is the horizontal component and is the vertical component

Part 5: Find the unit vector in the direction of each vector below. Verify that the unit vector has a magnitude of 1.

  1. 2.
  1. 4. 3.

Vector Computations

To add or subtract vectors, we add or subtract (in order) the components of the vectors.

To multiply a vector by a scalar, we multiply each component of the vector by the scalar.

Part 6: Perform each operation belowgiven the picture on the right.

  1. 2. 3.

Part 7: Given each vector below, find the magnitude and direction of the resultant vector.

  1. 2.

Part 8: Complete each problem below.

  1. A pilot flies a plane due west for 150 miles, then turns 42° north of west for 70 miles.

Find the plane’s resultant distance and direction from the starting point.

  1. A ferry shuttles people from one side of a river to the other. The speed of the ferry in still water is 25 mi/h. The river flows directly north at 9 mi/h. If the ferry heads directly west, what are the ferry's resultant speed and direction?

Resulting speed = ______

Describe the direction (include angle and compass direction):

  1. To find the distance between two points A and B on opposite sides of a lake, a surveyor chooses a point C which is 720 feet from A and 190 feet from B. If the angle at C measures 68, find the distance from A to B.
  1. A baseball is thrown at a 22.5 angle with an initial velocity of 70 m/s. Assume no air resistance and remember that the acceleration due to gravity is .
  1. What is the initial vertical component of the ball’s velocity?
  1. What is the horizontal component of the ball’s velocity?
  1. How long until the ball hits the ground?
  1. How high did the ball travel?
  1. How far did the ball travel horizontally when it hit the ground?
  1. Without the wind, a plane would fly due east at a rate of 150 mph. The wind is blowing southeast at a rate of 50 mph. The wind is blowing at a 45° angle from due east. How far off of the due east path does the wind blow the plane?

Homework

Part 1: Use the given vectors to sketch each vector below.

  1. 2. 3. 4. 5.

Part 2: Find and the direction angle

  1. 2.

Part 3: Given that and , perform each operation below.

  1. 2. 3. 4.

Part 4: Complete the problem below.

  1. A plane flies due west at 250 kilometers per hour while the wind blows south at 70 kilometers per hour. Find the plane’s resultant velocity and bearing.
  1. A plane flies east for 200 kilometers, then east of south for 80 kilometers. Find the plane’s distance travelled and bearing from its starting point.
  1. A plane is heading west of south. After 240 miles the pilot changes his direction to west of south. After he has travelled 480 further, find the distance and direction angle from its starting point.
  1. A clown is shot out of a cannon with a velocity of 200 feet per second at an angle of with the horizontal. Find the vertical and horizontal components of the velocity of this clown.