Geometric and Algebraic

A vector is a quantity that has both magnitude and direction.

We use an arrow to represent a vector. The length of the arrow is the magnitude

and the arrowhead indicates the direction of the vector.

The points on a line joining the two points form what is called a line segment.

If we order the points so that they proceed from one point to another, we have a directed line segment or a geometric vector, denoted by the two points with an arrow on top of it.

The beginning point is called the initial point. The ending point is called the terminal point.

The vector v whose magnitude is 0 is called the zero vector, 0. The zero vector is assigned no direction.

Two vectors are equal if they have the same magnitude and direction. (Both must be the same.)

See example on p. 619.

To find the sum of two vectors we position the vectors so that the terminal point of the first vector coincides with the initial point of the second vector.

Vector addition is commutative v + w = w + v

Vector addition is also associativeu + (v + w) = (u + v) + w

The zero vector has the property that v + 0 = 0 + v = v

If v is a vector, then –v is the vector having the same magnitude as v, but whose direction is opposite to v.

v + (-v) = 0

If is a scalar and v is a vector, the scalar product v is defined as follows:

1. If > 0, the product v is the vector whose magnitude is times the magnitude of v and whose direction is the same as v.

2. If <0, the product v is the vector whose magnitude is |times the magnitude of v and whose direction is the opposite that of v.

3. If a = 0 or if v = 0, then v = 0.

0v = 0 1v = v -1v = -v

(v = v + v (v = v + v

v)=(v

Graphing Vectors p. 628 #s 2 – 8 even

Magnitudes of Vectors If v is a vector, we use the symbol ||v|| to

represent the magnitude. ||v|| is the length of a directed line segment.

Properties of Vectors If v is a vector and if is a scalar, then

(a) ||v|| ≥ 0 (b) ||v|| = 0 iff v = 0

(c) ||-v|| = ||v|| (d) ||v|| = || ||v||

Unit Vector A vector u for which the magnitude of

vector u is equal to one is called a unit vector.

Algebraic Vectors An algebraic vector is represented by an

ordered pair written in brackets.

,a b

where a and b are real numbers (scalars) called the components of the vector v.

Position Vector

1 1, 1

2 2 2 1 2

Suppose that v is a vector with initial point P

not necessarily the origin, and terminal point

P , . If , then is equal to the

position vector

x y

x y v PP v

88888888888888888888888888 88

2 1 2 1,v x x y y

Finding a Position Vector Find the position vector of the vector

1 2 1 2, if 2, 1 ; 6, 2 .v PP P P 8888888888888 8

6 ( 2) , 2 ( 1)

8, 1

v

Equality of Vectors Two vectors v and w are equal iff their

corresponding components are equal.

1 1 2 2

1 2 1 2

If , and ,

then

v a b w a b

v w iff a a and b b

8888888888888 8

8888888888888 8

We call a and b the horizontal and vertical components of v, respectively.

Unit Vectors

Let denote the unit vector whose direction is

along the positive x-axis; let denote the unit vector

whose direction is along the positive y-axis.

Then 1,0 and 0,1 . Any vector can be

written u

i

j

i j

88888888888888

sing the unit vectors .i and j

Addition and Subtraction Using Unit Vectors

1 1 1 1 2 2 2 2

1 2 1 2 1 2 1 2

1 2 1 2 1 2 1 2

Let , and ,

be two vectors, and let be a scalar. Then

,

,

v a i b j a b w a i b j a b

v w a a i b b j a a b b

v w a a i b b j a a b b

888888888888888888888888888888888888888 888

888888888888888888888888888888888888888 888

888888888888888888888888888888888888888 888

Scalar Product and Magnitude Using Unit Vectors

1 1 1 1

2 21 1

,v a i b j a b

v a b

Unit Vector in the Direction of v For any nonzero vector v, the vector

is a unit vector that has the same direction as v.

vu

v

Finding a Unit Vector Find a unit vector in the same direction as v = 2i – j We find ||v|| first.

222 1 5v

We now divide the original vector by the

magnitude of the vector.

2 1

5 5i j

Writing a Vector in Terms of Its Magnitude and Direction If a vector represents the speed and

direction of an object, it is called a velocity vector.

If a vector represents the direction and amount of a force acting on an object, it is called a force vector.

The following helps to find the velocity and force vectors.

Vector in Terms of Magnitude and Direction A vector written in terms of magnitude and

direction is

v = ||v|| (cos i + sin j)

where is the angle between v and i.

Writing a Vector when Magnitude and Direction Are Given A child pulls a wagon with a force of 40

pounds. The handle of the wagon makes an angle of 30o with the ground. Express the force vector F in terms of i and j.

The magnitude of the Force is given as 40 pounds.

Now express the vector in terms of magnitude and direction

Force Problem Continued

40 cos30 sin 30

3 140 20 3 20

2 2

The horizontal component of the force is 20 3 and

the vertical component is 20.

o oF i j

i j i j

888888888888888888888888888888888888888888

Static Equilibrium If two forces simultaneously act on an

object the vector sum is known as the resultant force. An application of this concept is static equilibrium. An object is said to be in static equilibrium if (1) the object is at rest and (2) the sum of all forces acting on the object is zero.

Object in Static Equilibrium A weight of 1000 pounds is suspended from

two cables as shown in the figure on p. 629 problem 59. What is the tension of the two cables.

Object in Static Equilibrium On-line Example