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VECTORS - PowerPoint PPT Presentation


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VECTORS. 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. Blue and orange vectors have same magnitude but different direction.

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PowerPoint Slideshow about 'VECTORS' - corby


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Presentation Transcript
slide2

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.

Blue and orange vectors have same magnitude but different direction.

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

Blue and greenvectors have same direction but different magnitude.

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

slide3

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

Q

The distance formula

Terminal Point

magnitude is the length

direction is this angle

Initial Point

P

How can we find the direction? (Is this all looking familiar for each application? You can make a right triangle and use trig to get the angle!)

slide4

Although it is possible to do this for any initial and terminal points, since vectors are equal as long as the direction and magnitude are the same, it is easiest to find a vector with initial point at the origin and terminal point (x, y).

Q

Terminal Point

A vector whose initial point is the origin is called a position vector

direction is this angle

Initial Point

P

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

slide5

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).

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).

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

Move w over keeping the magnitude and direction the same.

Initial point of v

slide6

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

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

slide8

Vectors Worksheet #1

Head-Minus Tail Rule

Prove that the two vectors RS and PQ are equivalent.

1.) R = (-4, 7) S = (-1, 5) P = (0, 0) Q = (3, -2)

2.) R = (7, -3) S = (4, -5) P = (0, 0) Q = (-3, -2)

3.) R = (2, 1) S = (0, -1) P = (1, 4) Q = (-1, 2)

4.) R = (-2, -1) S = (2, 4) P = (-3, -1) Q = (1, 4)

slide9

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.

Vectors are denoted with bold letters

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.

(a, b)

(3, 2)

slide10

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

When we want to know the magnitude of the vector (remember this is the length) we denote it

Let's look at this geometrically:

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

slide11

Vectors Worksheet #2

Using the following vectors:

P = (-2, 2) Q = (3, 4) R = (-2, 5) S = (2, -8)

Find:

PQ RS QR PS

2QS (√2)PR 3QR + PS PS – 3PQ

slide12

Vector Worksheet #3

Performing Vector Operations

Let u = <-1, 3> and v = <4, 7>

Find the component form of the following vectors.

u + v 3u 2u + (-1)v

u + v = <-1, 3> + <4, 7> = <-1 + 4, 3 + 7> = <3, 10>

3u = 3<-1, 3> = <-3, 9>

2u + (-1)v = 2<-1, 3> + (-1)<4, 7> = <-2, 6> + <-4, -7> = <-6, -1>

slide13

Performing Vector Operations

Now it’s your turn!

Let u = <-1, 3> , v = <2, 4> and w = <2, -5>

Find:

u + v

u + (-1)v

u – w

3v

2u + 3w

2u – 4v

– 2u – 3v

– u – v

slide14

Performing Vector Operations

Now it’s your turn!

Let u = <-1, 3> , v = <2, 4> and w = <2, -5>

Find:

u + v = <1, 7>

u + (-1)v = <-3, -1>

u – w = <-3, 8>

3v = <6, 12>

2u + 3w = <4, -9>

2u – 4v = <-10, -10>

– 2u – 3v = <-4, -18>

– u – v = <-1, -7>

unit vectors and direction angles
Unit Vectors and Direction Angles
  • Any vector can be broken down into its components: a horizontal component and a vertical component.
  • In addition, any vector can be written as an expression in terms of a standard unit vector.
  • Unit vectors help us separate vectors into components—a scalar and a unit vector.
unit vectors and direction angles1
Unit Vectors and Direction Angles
  • The standard unit vectors are i and j.
  • i = <1, 0>
  • j = <0, 1>
  • Using this style, we can now express vectors as a linear combination.
unit vectors and direction angles2
Unit Vectors and Direction Angles
  • Vector v = <a, b>

= <a, 0> + <0, b>

= a<1, 0> + b<0, 1>

  • We now have linear combination.

v = a•i + b•j

unit vectors and direction angles3
Unit Vectors and Direction Angles

In vector v = a•i + b•j

a and b are now scalars and express the horizontal and vertical components of vector v.

We can use Trigonometry to calculate a direction angle for our vector.

slide19

A unit vector is a vector with magnitude 1.

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.

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

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

slide20

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

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

slide21

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 Corcoran

Head of Mathematics

St Stephen’s School – Carramar

www.ststephens.wa.edu.au