Vector Mathematics AP

8/8/2019 Vector Mathematics AP

http://slidepdf.com/reader/full/vector-mathematics-ap 1/31

Vector Mathematics

Adding, Subtracting, Multiplying

and Dividing



Why?

One can add 23 kg and 42 kg and get 65

kg.

However, one cannot add together 23 m/ssouth and 42 m/s southeast and get 65

m/s south-southeast.

Vectors addition takes into account adding

both magnitude and direction



Words

Vector : A measured quantity with both

magnitude (the how big part) and direction

Scalar : A measured quantity withmagnitude only

Resultant Vector : The final vector of a

vector math problem



³Math´ Coordinate System

(Direction)

0º

90º

180º

270º



Polar Coordinate System

(Direction and Magnitude)





Polar ³Math´ (Cartesian)

2 2

1

cossin

tan

r x y

x r

y r

y

x

UU

U

!

!!

¨ ¸! © ¹ª º

x

yr



Vector addition

Two Ways:

1. Graphically: Draw vectors to scale, Tip

to Tail, and the resultant is the straightline from start to finish

2. Mathematically: Employ vector math

analysis to solve for the resultant

vector



Graphically 2-D Right

A = 5.0 m @ 0°

B = 5.0 m @ 90°

Solve A + B

R

Start

R=7.1 m @ 45°



Important

You can add vectors in any order and yield

the same resultant.







Let¶s add the last one

mathematically The math you used previously doesn¶t

work (and I won¶t let you use the Law of

Sines or Cosines) or does it???

What we will do is break each vector into

components

The components are the x and y values of

the polar coordinate (go back 6 slides)

Check out the next slides«







Components of Vectors

A = Ax + Ay

Ax =A cos

Ay = A sin

As long as you

draw the xcomponent first

A

Ax

Ay



The Table Method

We will organize these components in a

table.

See the board for this part and next slide



Table Method Equation

Add all X components together Final Rx

Add all Y components together Final Ry



Subtracting Vectors

Simply add or subtract 180° (keep

between 0° and 360°) to the direction of

the vector being subtracted

You just ADD the OPPOSITE vector (there

is no subtraction in vector math)



Subtracting Vectors



Unit Vectors

A unit vector is a vector that has a

magnitude of 1, with no units.

Its only purpose is to point We will use i, j, k f or our unit vectors

i means x ± dir ection, j is y, and k is z

We also put little ³hats´ (^) on i, j, k to show that they ar e unit vectors (I will

boldf ace them)



Unit Vectors for vectors A & B



Unit Vectors



Adding using unit vectors

R = A + B

R = (Ax + Bx )i + (Ay + By ) j + (Az + Bz )k

which becomesR = R x i + R y j + R z k

The magnitude of R is f ound by a pplying the

Pythagor ean theor em



Multiplying Vectors (products)

3 ways

1. Scalar x Vector = Vector w/

magnitude multiplied by the value of

scalar

A = 5 m @ 30°

3A = 15m @ 30°




2. (vector) (vector) = Scalar

This is called the Scalar Product or the

Dot Product



Dot Product Continued (see p. 25)

A

B




3. (vector) x (vector) = vector

This is called the vector product or the

cross product



Cross Product Continued



Cross Product Direction and

reverse



Cross Product

You can also solve the Cross Product with

a matrix and unit vectors«check out the

board for this.

Documents

Vector Mathematics AP