Vectors
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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.

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Vectors

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
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
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 vectors1
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
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
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
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
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
Add the following

  • d1 = 30m, 60o East of North

  • d2 = 20m, 190o

    • dr = d1+d2

    • dr = ?

    • dr = 13.1m, 61o


Vector subtraction
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
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
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 this1
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
Now try this:

VX = 25m/s

VY = - 51m/s

Find V=


And your point is
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 example1
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 = ???


A review of an example
A Review of an Example

y (km)

x (km)


But wait there s more
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
Practice it:

  • Pg. 70, # 1, 4


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