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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)

90º

180º

0º

270º

Polar Coordinate System (Direction and Magnitude)

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Polar ?? Math (Cartesian)

r

y

?

x

Vector addition

- Two Ways
- Graphically Draw vectors to scale, Tip to Tail,

and the resultant is the straight line from start

to finish - Mathematically Employ vector math analysis to

solve for the resultant vector

Graphically 2-D Right

- A 5.0 m _at_ 0
- B 5.0 m _at_ 90
- Solve A B

R

R7.1 m _at_ 45

Start

Important

- You can add vectors in any order and yield the

same resultant.

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Lets add the last one mathematically

- The math you used previously doesnt work (and I

wont 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 6s) - Check out the nexts

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Components of Vectors

- A Ax Ay
- Ax A cos ?
- Ay A sin ?
- As long as you draw the x component first

A

Ay

?

Ax

The Table Method

- We will organize these components in a table.
- See the board for this part and next

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 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 )kwhich

becomes R Rx i Ry j Rz k - The magnitude of R is found by applying the

Pythagorean theorem

Multiplying Vectors (products)3 ways

- Scalar x Vector Vector w/ magnitude multiplied

by the value of scalar A 5 m _at_ 303A 15m

_at_ 30

Multiplying Vectors (products)

- 2. (vector) (vector) Scalar
- This is called the Scalar Product or the Dot

Product

Dot Product Continued (see p. 25)

B

F

A

Multiplying Vectors (products)

- (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 vectorscheck out the board for

this.

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