advertisement

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