Vectors

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# Vectors - PowerPoint PPT Presentation

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.

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

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Presentation Transcript

### Vectors

You will be tested on your ability to:

correctly express a vector as a magnitude and a direction

break vectors into their components

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?
• 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
• 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

Now try this:

VX = 25m/s

VY = - 51m/s

Find V=

• 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