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

,
1 2 of real numbers.
 a a
1 2
 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

,
1 2
and the position vector of P
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

,
1 2
 y
1
3.
(a) Draw the points A
 
and B

2, 1

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

,
1 2
 y
1
. a

Length of a Vector The length of a vector 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

,
1 2 a

, write down the formula for a .
(f) Given a 3 dimensional vector a

, ,
1 2 3 a

, guess the formula for a .
(g) Given a

3, 2, 1 , compute a . a

4

Vector Addition
Let a

,
1 2
and b

,
1 2
, the sum of the two vectors is defined as a b ,
1 2

,
1 2
 a
1

,
1 2
 b
2
Scalar Multiplication
Let 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
 v and 2 v .
(b) Draw u , v , u
 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

,
1 2
, b

,
1 2
,
  b
 a
1

,
1 2
 b
2
6. a

1, 2,3 , b
  a

2 b

Length Formula Let a

,
1 2
and c a scalar. A natural question is, “What is the relationship between the lengths of 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 stands for the collection of the n -dimensional vectors. n
Zero Vector In V , what is the zero vector in properties 3 and 4?
2
0

,
Standard Basic Vectors
V
2
: i

1, 0 , j

V i
3
:

1, 0, 0 , j

0,1
0,1, 0 , k

0, 0,1
8. Draw the standard basic vectors i

1, 0 , j

0,1 on the graph paper.
7

9. (a) Express a

1, 2,3 j

0,1, 0 , and k

0, 0,1
and
.
b

1, 0, 1 in terms of the basic vectors i

1, 0, 0 , a

1, 2,3
 b

1, 0, 1
(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.
Unit Vector u in the Same Direction as Another Vector b : u
 b

1 b 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

8

Classwork
1. a
 
2 j

,
  
2 j
 k
(a) Compute a

2 b . 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

1 a

2 b
 a

2 b
 
9