Chapter 2 - Vectors

Objectives• Understand vectors and their

components on the coordinate system

• Find the magnitude of a vector• Understand vector addition by

the parallelogram method and by the component method

• Understand vectors in a state of equilibrium

Vectors on the Coordinate System

• Horizontal, vertical, and slanted vectors can be drawn on the coordinate system.

• All 3 types of vectors have both length and direction.

Slanted Vectors

• The direction of slanted vectors is stated in terms of – The angle formed by the vector and the

horizontal axis. – The quadrant in which that angle is

formed.

• The length of this vector is ___?• The direction of this vector is a ___

angle in the ___ quadrant.3 units

Θ = 40˚

Slanted Vectors

• The angle which specifies the direction of a slanted vector is called its reference angle.

• All slanted vectors have positive lengths.

• Vectors are named using 2 letters:– AB

• The first letter of the name is always where the vector begins.

3 units

Θ = 40˚A

B

Slanted Vectors

• Any slanted vector has a horizontal and a vertical component.

• We can calculate these because we can make this a right triangle and use trig.

3 units

Θ = 40˚A

B

Magnitude

• How long is this vector?• Use the distance formula!

• If A(x1, y1) and B(x2, y2) are points in a coordinate plane, then the distance between A and B is

• AB = √(x2 - x1)2 + (y2 - y1)2

• AB = (4-0)2 + (5-0)2

• AB = √ 16 + 25• AB = √41

4,5

0,0A

B

Magnitude

• Try another one:

• AB = √(x2 - x1)2 + (y2 - y1)2

• AB = √(5-0)2 + (4-2)2

• AB2 = √ 25 + 4• AB = √29 = 5.4

5,4

0,2 A

B

Component Form

• The component form of a vector is written as <x, y> where x is (x2 - x1) and y is y2 – y1

• What is the component form of this vector?

• <(5-0), (4-2)>• <5, 2>

5,4

0,2 A

B

Another example

• What is the component form of this vector?

• <4, 5>

4,5

0,0A

B

Direction

• The direction of a vector is determined by the angle it makes with the horizontal line.

• What direction is vector AB heading?• If AB represents the velocity of a

moving ship, and the scale on the axis is miles per hour, how fast is the ship moving? 3,4

0,0A

B

Equal and Parallel Vectors

• Two vectors are equal if they have the same magnitude and direction.

• Two vectors are parallel if they have the same or opposite directions.

3,4

0,0A

B

Slanted Vectors

• How do we calculate the horizontal component (AC)?

• Cos θ = adj/hyp = x/3• .7660 = x/3• X = 3 * .7660 = 2.298

• sin θ = opp/hyp = x/3• .6428 = x/3• X = 3 * .7660 = 1.9284

• Use Pyth to check• 2.2982 + 1.92842 ?=? 32

3 units

Θ = 40˚A

B

C

Flipping the problem

• Tan = opp/adj• Tan θ = 4/5 = .8000• Therefore θ contains 39˚

• Pyth can help us find the length of AB:• AB2 = AC2 +BC2 • AB = 52 + 42

• AB = 25 + 16 = 41 • AB = 6.4

• How would you do this using sin and cos?

Θ = ?˚A

B = (5, 4)

C

Adding Vectors• What does it mean to add two

vectors?• Vector and Field (vector addition)• Why do we care? Using Vectors

Video

A= 5,20

C = -4,3θ

Adding Vectors

• In Physics, the Law of Conservation and Momentum uses this.

• Now how do we do that without the website?

• Create a parallelogram and find the diagonal.

A= 5,20

C = -4,3θ

Adding Vectors• Draw AQ which is both parallel

to OC and equal in length to OC.

• Draw CQ which is both parallel to OA and equal in length to OA

• On a graph, we can see that the points of Q are 2,6

A= 5,20

C = -4,3θ

Q

Adding Vectors• We can draw one line, then a vector

from the origin to point Q:• This lets us find the point on graph

paper without a calculator. Even using a calculator, this is a nice way to prove we’re doing things correctly.

• Would this be precise if we weren’t using whole numbers?

A= 5,20

C = -4,3θ

Q

Adding Vectors• Another way to add vectors is

by the component method.– This provides accurate answers

without the necessity of constructing parallelograms.

• Find the horizontal and vertical components, and add them

• Horizontal: -3 + 5 =2• Vertical: 4 + 2 = 6

A= 5,20

C = -3, 4θ

Q

Vector addition• Positives and negatives are

extremely important – be careful with them.

A= 5,20

C = -4,3θ

Q

Vector addition• To find the length and direction

of the resultant vector, we use trig.

• Use Pyth to find the length of OC

• Use tan to find the reference angle of OC

C= 25.6, 12.70

F = -7.9, 7.2

α

Q

Applications

• How is vector addition used in physics?

• Law of Conservation Video