Geom9point7

Chapter 2 - Vectors

Objectives• Understand vectors and their

components on the coordinate system

• 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.

• All 3 types can be represented by signed numbers.

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

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

Try some of these• Do p. 81 #s 54-62

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? Canoe

A= 5,20

C = -4,3θ

QuickTime™ and aSorenson Video 3 decompressorare needed to see this picture.

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?

QuickTime™ and aSorenson Video decompressorare needed to see this picture.

Homework• Do page 576 10-30 evens

Adding Vectors on the Coordinate Axes

• On page 107, #103, the vertical components are both– 0– And the reference angle is– 0˚

• So adding these is pretty easy!• On #104, the horizontal

components are both– 0– And the reference angle is– 90˚

• Once again, addition is easy

Finding a Vector-Addend• The sum of a vector addition is

called the resultant. • The two vectors which are

added are called vector addends.

• Use Pythagorean and tan

Adding more than two vectors• The parallelogram method

doesn’t work – use the component method

• See example at the bottom of page 116.

Vector Opposites, the Zero-Vector, and the State of Equilibrium

• Two numbers are a pair of opposites if their sum is 0.

• Two vectors are a pair of vector opposites if both – The sum of their horizontal components

is 0– The sum of their vertical components is 0

• See example on p. 119 #122• When the resultant of two vectors is

the zero vector, we say that the two vectors are in a state of equilibrium.

• Can you think of an example of this in real life?

The Zero-Vector

• What is the horizontal component of the zero vector?– 0

• What is the vertical component of the zero vector?– 0

• What is the length of the zero vector?– 0

Equilibrants and the State of Equilibrium

• If a system of vectors is not in a state of equilibrium, we can always add one more vector which produces a state of equilibrium. This added vector is called an equilibrant.

• If 3 vectors are in a state of equilibrium, each vector is the vector-opposite of the resultant of the other 2 vectors.

• See examples on p. 123

Homework• Do Self-Tests 9, 10, 11, 12 & 13