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## PowerPoint Slideshow about ' Vectors' - shea-buckley

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### Vectors

You will be tested on your ability to:

correctly express a vector as a magnitude and a direction

break vectors into their components

add and multiply vectors

apply concepts of vectors to linear motion equations (ch. 2)

Vector vs. Scalar

- Scalar units are any measurement that can be expressed as only a magnitude (number and units)
- Examples:
- 14 girls
- $85
- 65 mph
- Vector quantities are measurements that have BOTH a magnitude and direction.
- Examples:
- Position
- Displacement
- Velocity
- Acceleration
- Force

Representing Vectors

- Graphically, vectors are represented by arrows, which have both a length (their magnitude) and a direction they point.

v = 45 m/s

d = 50 m

v = 25 m/s

a= 9.8 m/s2

Representing Vectors

- The symbol for a vector is a bold letter.
- For velocity vectors we write v
- For handwritten work we use the letter with an arrow above it. v
- Algebraically
- Vectors are written as a magnitude and direction
- v = lvl , Θ
- Example v = 25 m/s, 120o or d = 50 m, 90o

Drawing Vectors

- Choose a scale
- Measure the direction of the vector starting with east as 0 degrees.
- Draw an arrow to scale to represent the vector in the given direction
- Try it! v = 25 m/s, 190o scale: 1 cm = 5 m/s
- This can be described 2 other ways
- v = 25 m/s, 10o south of west
- v = 25 m/s, 80o west of south
- Try d = 50 m, 290o new scale?

Adding Vectors

- Vector Equation
- vr = v1+v2
- Resultant- the vector sum of two or more vector quantities.
- Numbers cannot be added if the vectors are not along the same line because of direction!
- Example……
- To add vector quantities that are not along the same line, you must use a different method…

An Example

D3

D2

DT

D1

D1 = 169 km @ 90 degrees (North)

D3 = 195 km @ 0 degrees (East)

D2 = 171 km @ 40 degrees North of East

DT = ???

Tip to tail graphical vector addition

- On a diagram draw one of the vectors to scale and label it.
- Next draw the second vector to scale, starting at the tip of the last vector as your new origin.
- Repeat for any additional vectors
- The arrow drawn from the tail of the first vector to the tip of the last represents the resultant vector
- Measure the resultant

Add the following

- d1 = 30m, 60o East of North
- d2 = 20m, 190o
- dr = d1+d2
- dr = ?
- dr = 13.1m, 61o

Vector Subtraction

- Given a vector v, we define –v to be the same magnitude but in the opposite direction (180 degree difference)
- We can now define vector subtraction as a special case of vector addition.
- v2 – v1 = v2 + (-v1)
- Try this
- d1 = 25m/s, 40o West of North
- d2 = 15m/s, 10o
- 1cm = 5m/s
- Find : dr = d1+d2
- Find dr = d1- d2

v

–v

Multiplying a vector v by a scalar quantity c gives you a vector that is c times greater in the same direction or in the opposite direction if the scalar is negative.

cV

V

-cV

Vector Components

- A vector quantity is represented by an arrow.
- v = 25 m/s, 60o
- This single vector can also be represented by the sum of two other vectors called the components of the original.

v = 50 m/s, 60o

sinΘ = Vy / V

Vy= V sinΘ

cosΘ = Vx / V

Vx= V cosΘ

Try this:

V2 = 10 m @ 30 degrees above –x

Find: V2X =

V2Y =

V1 = 10 m @ 30 degrees above +x

Find: V1X =

V1Y =

Ө2

Ө1

But V2X should be NEGATIVE!!!

Try using the angle 150 degrees for V2

Try this:

V2 = 10 m @ 30 degrees above –x

Find: V2X =

V2Y =

V1 = 10 m @ 30 degrees above +x

Find: V1X =

V1Y =

Find : V3X =

V3Y=

V3 = 10 m @ 30 degrees below +x

Try using the angle 330 degrees for V3

and your point is???

- ALWAYS:

Describe a vector’s direction relative to the +x axis

- ALWAYS:

Measure counter-clockwise angles as positive

- ALWAYS:

Measure clockwise angles as negative

An Example

D3

D3X

D2

D2Y

D2X

DT

D1Y

D1

D1 = 169 km @ 90 degrees (North)

D3 = 195 km @ 0 degrees (East)

D2 = 171 km @ 40 degrees North of East

DT = ???

But Wait. . . There’s more!

We’ve Found:

DTX = 326 km

DTY = 279 km.

y (km)

For IDTI, use the Pythagorean Theorem.

For the Direction of DT, use Tan-1

x (km)

Practice it:

- Pg. 70, # 1, 4

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