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/s south 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 with magnitude only

• Resultant Vector: The final vector of a vector math problem

“Math” Coordinate System (Direction)

0º

90º

180º

270º

Polar Coordinate System (Direction and Magnitude)

Polar “Math” (Cartesian)

2 2

1

cossin

tan

r x yx ry r

yx

x

yr

θ

Vector addition

• Two Ways:1. Graphically: Draw vectors to scale, Tip

to Tail, and the resultant is the straight line 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 θ

• Ay = A sin θ

• As long as you draw the x component first

A

Ax

Ay

θ

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 θ 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 for our unit vectors• i means x – direction, j is y, and k is z• We also put little “hats” (^) on i, j, k to show

that they are unit vectors (I will boldface 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 becomes R = Rx i + Ry j + Rz k

• The magnitude of R is found by applying the Pythagorean theorem

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°

Multiplying Vectors (products)

2. (vector) • (vector) = ScalarThis is called the Scalar Product or the

Dot Product

Dot Product Continued (see p. 25)

Φ

A

B

Multiplying Vectors (products)

3. (vector) x (vector) = vectorThis 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.