vectors. what is the difference between velocity and speed? -speed is only magnitude -velocity is...

VECTOR

S

What is the difference between velocity and speed?

-Speed is only magnitude

-Velocity is magnitude and direction

ScalarWe use scalars (real numbers) to denote magnitude

MAGNITUDE like speed and mass

VectorsVectors have magnitude and directionThese denote quantities like velocity and force

A vector is a quantity that has both magnitude and direction. It is represented by an arrow. The length of the vector represents the magnitude and the arrow indicates the direction of the vector.

Two vectors are equal if they have the same direction and magnitude (length).

Blue and orange vectors have same magnitude but different direction.

Blue and green vectors have same direction but different magnitude.

Blue and purple vectors have same magnitude and direction so they are equal.

On the graph, plot a Vector that has an Initial point at (1,1)And a terminal point at (-7,6)

Plot another starting At (2,3) and endingAt (5, 7)

Equal vectors can be Translated to the origin To make them POSTION VECTORS

Translate the two vectorsfrom the last slide so that they are position vectors (at the origin)

When vectors are Position Vectors, we can write them in component form:u = < 3,4 > and v = < -8,5 >

PLOT v = < 7, 9 > and u = < -3, -6 >

If we subtract the initial point from the terminal point, we will have an equivalent vector with initial point at the origin.

Horizontal Component-

Vertical Component

What is the difference between distance and displacement?

To find out…….City Walk!

DRILL 9/2/14 *Take out syllabus if not turned in *Take out graphs from last class

Please copy these and multiply/distribute:

1.(x + 3)(3x – 5) =

2.(4x + 9)(-6x – 10) =

P

Q

Initial Point

Terminal Point

magnitu

de is th

e length

direction is

this angle

How can we find the magnitude if we have the initial point and the terminal point?

22 , yx

11, yx

The distance formula

Let’s go back and find the MAGNITUDE of the vectors on our graph paper -- what is the distance formula?

22 , yx

11, yx

Q

Terminal Point

Although it is possible to do this for any initial and terminal points, it is easiest to find the magnitude of a position vector.

ba,

0,0P

Initial Point

a

b

Let’s practice a few ….. What is the magnitude of v= <5, 7>?What about u = <-2, -5>?

Note: If time allows, the Gizmo activity will allow students to discover adding vectors algebraically (as per pacing guide).

We can complete vector operations by drawing and using algebraic operations.

Let’s start by drawing- Patty paper time!

Copy u = < 3,4 > and v = < -8, 5 > onto patty paper.

To add vectors, we put the initial point of the second vector on the terminal point of the first vector. The resultant vector has an initial point at the initial point of the first vector and a terminal point at the terminal point of the second vector (see below--better shown than put in words).

v w

Initial point of v v

w

Move w over keeping the magnitude and direction the same.

To add vectors, we put the initial point of the second vector on the terminal point of the first vector. The resultant vector has an initial point at the initial point of the first vector and a terminal point at the terminal point of the second vector (see below--better shown than put in words).

Tip-to-tail: To add vectors, we put the initial point of the second vector on the terminal point of the first vector. The resultant vector has an initial point at the initial point of the first vector and a terminal point at the terminal point of the second vector (see below--better shown than put in words).

Terminal point of w

w

Using patty paper, find u + v

Is v + u the same?

v w

Initial point of v v

w

Parallelogram method: Sketch the initial points of the vectors at the same point. The sum v + w is the diagonal of the parallelogram formed by u and v.

w w

What about u - v? What would that look like?

On graph paper, find u + v using the parallelogram method.

Is it the same as the tip to tail method?

The negative of a vector is just a vector going the opposite way.

v

v

Try on your graph or patty paper. What does u – v look like?

We can also do vector operations algebraically.

Let u = < 2, -7 > and v = < -3, 4 >

Find u + v

Find u - v

Your turn: Let u = <1, 2> and v = <-5, -4>.

Find u + v.

Find u - v

A number multiplied in front of a vector is called a scalar. It means to take the vector and add together that many times.

vvv3

v

uv

w vu u

vw3

w ww

Using the vectors shown, find the following:

vu u

v

vwu 32

uu w

w wv

A few examples using scalars given u = <-5, 4> and v = <3, 2>

1.-3u

2.5(u-v)

3.2u - 3v + 4u

,a b v

Vectors are denoted with bold letters

(a, b)

This is the notation for a position vector. This means the point (a, b) is the terminal point and the initial point is the origin.

We use vectors that are only 1 unit long to build position vectors. i is a vector 1 unit long in the x direction and j is a vector 1 unit long in the y direction.

ij

,a b a b v i j

(3, 2)

ij

3,2 v i i

j

jiv 23

UNIT VECTORS

What is u = <5, -6 > written in unit vector form?

Try these:

v = < -3, 7 >

u = < -8, -3 >

jiji 4352

If we want to add vectors that are in the form ai + bj (unit vector form), we can just add the i components and then the j components.

jiv 52

wv ji

Let's look at this geometrically:

i2

j5 v

i3

j4w

ij

v 22 52 Can you see from this picture how to find the length of v?

29

jiw 43

Let’s practice a few operations in unit vector form:

(3i + 2j) + (5i – 2j)

(i – 2j) - (-4i + j)

(6i + 2j) – (7i – 4j)

A unit vector is a vector with magnitude 1.

jiw 43

If we want to find the unit vector having the same direction as a given vector, we find the magnitude of the vector and divide the vector by that value.

What is ?w

2 23 4 w 525

If we want to find the unit vector having the same direction as w we need to divide w by 5.

jiu5

4

5

3

Let's check this to see if it really is 1 unit long.

2 23 4 25

15 5 25

u

Let’s practice finding unit vectors in a given direction.

Find a unit vector in the same direction as v = < -3, -4 >

1: Find the magnitude of v

2: Divide the vector v by it’s magnitude (leave in fraction form).

3: Check: Your new unit vector should have a magnitude of 1.

YOUR TURN: Find a unit vector in the same direction as v = <5, -12>

Teachers: Conclude the lesson by completing the Vector Representations Summarizer addendum. In this addendum students are given a table including a plot, magnitude, initial/terminal points, component form, or unit vector form and asked to fill in the missing information. This could be modified to use in a number of classroom games.

Teachers: the next slide is ahead of where days 2-3 have us.I am still working on 4-5.

If we know the magnitude and direction of the vector, let's see if we can express the vector in ai + bj form.

5, 150 v

1505

As usual we can use the trig we know to find the length in the horizontal direction and in the vertical direction.

cos sin v v i j

jijiv2

5

2

35150sin150cos5

Acknowledgement

I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint.

www.slcc.edu

Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified to suit the Western Australian Mathematics Curriculum.

Stephen CorcoranHead of MathematicsSt Stephen’s School – Carramarwww.ststephens.wa.edu.au