Section 1 Objectives

The student should be able to:

1. Distinguish between a scalar and a vector

2. Combine vectors using graphical methods

3. Multiply and divide vectors by a scalar

ScalarsNeed to Know

• Specified by a magnitude and a unit– 4 m/s– 10 kg– 10 x 1012 m

VectorNeed to Know

• Specified by a magnitude and unit AND

DIRECTION– 4 m/s heading west– 10 x 1012 m north– 10 m/s2 down

• As long as the direction and magnitude are kept the same you can move the vector anywhere

Vector Representation Need to Know

• On a drawing, a vector is represented by an arrow

• The length of the vector is proportional to the magnitude

• In print, a vector is usually bold• In hand written work, a vector can be

indicated by an arrow over it

Vector Addition Need to Know

• If they are collinear, simple arithmetic can be used

• Simple arithmetic can not be used if they are not collinear

• The sum of a given set of vectors is

called the resultant

Example

Suppose you drive 200 km to the east and then 50 km to the west. What is your total displacement?

200 km east (+)

50 km west (-)

Since they are parallel I can add arithmeticallyI assume everything going to the right is positiveAnd everything going to the left is negative

Displacement = 200 km – 50 kmResultant = 150 km to east (+)

150 km east (+)

What if they are not collinear or parallel?

• We can add them together graphically– Tip to tail method– Parallelogram method

• We can add them together mathematically with trigonometry (oh my!)

Graphical Addition Need to Know

• Tip to Tail method:– Draw first vector to scale– Draw second vector to scale, placing its tail

at the first vector’s tip (make sure your directions are correct!)

– Draw an arrow from the tail of the first vector to the tip of the second vector. This is the resultant of the two vectors

– Approximate the length of the resultant

Tip to Tail Method

20 m

15 m

ResultantApproximately ≈ 25 m

20 m + 15 m

TailsLineup

TipsLineup

Tip to tail

Tip to Tail Method

+

20 m

20 m

10 m

10 m

Resultant ≈ 25 m

Website Example

Adding vectors tip to tail

Simulation

Tip to tail with numbers

+ =1010

Resultant

+ =1012 10

+

Multiplying Vectors by Scalars

• A vector can be multiplied (or divided) by a scalar

• Result is a vector

5

Graphical Addition Need to Know

• Parallelogram method– The tails of the vectors are drawn from a

common origin– Parallelogram is constructed using these

two vectors as adjacent sides– The resultant is drawn from the common

origin– We can only add two at a time with this

method

Parallelogram Method

+20 m

15 m

Tails aretogether 20 m

15 m

Createparallelogramwith oppositesides

≈ 23 m

Parallelogram Method

30

15≈35

+15

30

+ =

+ =

Graphical Addition

Bottom Line: Gives a good approximate direction and magnitude of the resultant vector.

For the most accurate results you must add your vectors mathematically!!

That is next ….. but first what do you recall about vectors

94 m/s is a1. Vector

2. Scalar

3. Direction

Correct answer is 2—scalar

94 m/s going west is a1. Vector

2. Scalar

3. Direction

Correct answer is 1--vector

A vector has1. Direction and

magnitude

2. Magnitude only

3. Direction only

Correct Answer is 1

The drawing indicates what type of vector addition?

1. Tip to tail

2. Parallelogram

Correct Answer is 2

The drawing indicates what type of vector addition?

1. Tip to tail

2. ParallelogramCorrect Answer is 1

Properties of Vectors

• Vectors can be moved parallel to themselves in a diagram

• Vectors can be added in any order• To subtract a vector, add its opposite

Practice

• Quest Vectors assignment