13.4 Vectors

• When a boat moves from point A to point B, it’s journey can be represented by drawing an arrow from A to B.

• AB

• Read vector AB

A

B

Vectors

• Vectors have– Direction– Magnitude (Length, Distance)

AB = (change in x, change in y)

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9-8-7-6-5-4-3-2-1

123456789

AB

AB = (change in x, change in y)

• Going from point A to point B– How much is there a change in the x direction?– How much is there a change in the y direction?

AB

AB = (5, 2)

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9-8-7-6-5-4-3-2-1

123456789

AB

Magnitude of vector AB

• |AB|

• The length of the arrow

• Use Pythagorean Theorem or the Distance formula.

Scalar Multiples

• 3 AB

Scalar Multiples

• -2 AB

Scalar Multiples

• -2 AB

White Board Practice

• Given points P(-3,4) and Q(-2,-2)a) Sketch PQ

b) Find PQ

c) Find |PQ|

d) Find 3PQ

e) Find -2PQ

White Board Practice

• Given points P(-1,-5) and Q(5,3)a) Sketch PQ

b) Find PQ

c) Find |PQ|

d) Find 3PQ

e) Find -2PQ

Equal Vectors

• 2 vectors are equal if they have the same magnitude and the same direction

Vector Sums

• To add 2 vectors

PQ + QR = PR

(4,1)+(2,3) = (6,4)

Definition

A vector is defined to be a directed line segment. It has both direction and magnitude (distance). It may be named by a bold-faced lower-case letter or by the two points forming it - the initial point and the terminal point. Examples: u or AB

uA

B

u

Equal VectorsTwo vectors are equal if they have the same

distance and direction.

u A

BAB=

Opposite VectorsOpposite vectors have the same magnitude, but opposite directions. That is, the terminal point of one is the initial point of the other.

u

-u

Resultant Vectors(adding)

When vectors are added or subtracted, the sums or differences are called resultant vectors.

Geometrically, we add vectors by placing the initial point of the second vector at the terminal point of the first vector in a parallel direction. The resultant vector has the initial point of vector 1 and the terminal point of the displaced vector 2.

A

B

C

D

A

BC

D

AB + CD = AD

Resultant Vectors(subtracting)

We subtract a vector the algebraic way by adding the opposite.

AB - CD = AB + (-CD)=AD

A

B

C

D

-(CD)

-(CD)

AD