vector formula

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MAT 1236 12.2 Discovery Lab

Objectives

To explore the vectors in the plane.

To investigate the 2 important operations: vector addition and scalar multiplication.

To understand some special vectors.

Instructions

Do not look up any references including the textbook and internet.

Use correct notations and do not skip steps.

Two persons per group.

Do not communicate with other groups.

Vectors

The term vector is used to indicate a quantity with both magnitude and direction.

Vectors in the Plane

A vector in the plane is represented graphically by a arrow from the origin to the point

a a

1

,

2

.

A two dimensional vector is an ordered pair

a

a a

1

,

2 of real numbers.

a a

1

,

2

a

a a

1

,

2

1

1.

Draw the vector

a

1, 2 on the graph paper.

Position Vector

Given any point

2.

Draw the point

P

,

2

,

OP

and the position vector of

a a

P

1

,

2

is the position vector of

on the graph paper.

P

.

2

Vectors From One Point to Another Point

Given 2 points can form a vector that points from

A

to

B

:

a

x

2

x y

1

,

2

y

1

3.

(a) Draw the points

A

 

and

B

 

on the graph paper.

1

,

1

 and

(b) Draw the vector

a

from point

A

to point

B

on the graph paper.

2

,

2

, we

(c) Compute the vector

a

by using the formula

a

x

2

x y

1

,

2

y

1

.

a

Length of a Vector

The length of a vector

a

a a

1

,

2 is denoted by

a

.

(d) Use a right angle triangle to compute the length of the vector

a

in part (b).

a

3

(e) Given

a

a a

1

,

2

, write down the formula for

a

.

a

(f) Given a 3 dimensional vector

a

1

,

2

,

3

, guess the formula for

a

.

a

(g) Given

a

, compute

a

.

a

4

Vector Addition

Let

a

a a

1

,

2

and

b

b b

1

,

2

, the sum of the two vectors is defined as

a a

1

,

2

b b

1

,

2

a

1

b a

1

,

2

b

2

Scalar Multiplication

Let

a

a a

1 2

and

c

a scalar, then the vector

ca

is defined as

ca

1

,

2

1

,

2

4. Let

u

5, 4 and

v

 

3, 2 .

(a) Compute

u

(b) Draw

u

,

v

,

u v

and 2

v

.

v

, and 2

v

on the graph paper. Make sure you label the vectors.

u v

2

v

5. Let

u

 

4, 6 and

v

2,3 .

Compute

1

2

u

v

the vectors.

and draw

u

,

v

, and

1

2

u

v

on the graph paper. Make sure you label

1

u

 

2

5

Vector Subtraction

One can define the vector subtraction in terms of vector addition..

a

a a

1

,

2

,

b

b b

1

,

2

,

a b a

 

b

a

1

b a

1

,

2

b

2

6.

a

1, 2,3 ,

b

a

2

b

Length Formula

Let

a

a a

1

,

2

and

c

relationship between the lengths of

a

a scalar. A natural question is, “What is the

a a

1 2

and

ca

ca ca

1 2

?”

7. (a) Let

a

1, 2 . Compute 2

a

,

a

, and 2

a

.

2

a

a

2

a

(b) Let

a

1, 2 . Compute

a

, and

a

.

 

a

 

a

(c) Based on the results on (b) and (c), guess a formula that relates

ca

and

a

.

ca

6

Properties

Note that

V n

stands for the collection of the

n

-dimensional vectors.

Zero Vector

In

V

2

, what is the zero vector in properties 3 and 4?

0

,

Standard Basic Vectors

V

2

:

i

V i

3

:

1, 0 ,

j

1, 0, 0 ,

j

8. Draw the standard basic vectors

i

0,1

0,1, 0 ,

k

1, 0 ,

j

0,1

0, 0,1

on the graph paper.

7

9. (a) Express

a

j

0,1, 0 , and

k

1, 2,3

0, 0,1

and

.

b

a

1, 2,3

b

1, 0, 1

in terms of the basic vectors

i

(b) Compute

a

2

b

by using the expressions of

a

and

b

in part (a).

a

2

b

Unit Vectors

A unit vector is a vector whose length is 1.

b

Unit Vector

u

in the Same Direction as Another Vector

b

:

u

b

1

b b

10. (a) Find a unit vector in the same direction of

b

6,8 .

(b) Draw

u

, and

b

on the graph paper. Make sure you label the vectors.

b

6,8

u

1

b b

1

6,8

6,8

1, 0, 0 ,

8

Classwork

1.

a

2

j

(a) Compute

a

2

b

.

2

j

k a

2

b

a

2

b

(b) Find a unit vector

u

in the same direction of

a

2

b

in terms of the basic vectors.

u

a

1

2

b

a

2

b

9

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