Vectors

Vectors

You will be tested on your ability to:1. correctly express a vector as a

magnitude and a direction2. break vectors into their components

3. add and multiply vectors4. 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

v = 25 m/s d = 50 m

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

D1

D2

D3

DT

D1 = 169 km @ 90 degrees (North)

D2 = 171 km @ 40 degrees North of East

D3 = 195 km @ 0 degrees (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.

V

cV

-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, 6

0o

sinΘ = Vy / V

Vy= V sinΘ

cosΘ = Vx / V

Vx= V cosΘ

Try this:V1 = 10 m @ 30 degrees above +x

Find: V1X = V1Y =

V2 = 10 m @ 30 degrees above –x

Find: V2X = V2Y =

But V2X should be

NEGATIVE!!!

Try using the angle 150 degrees for V2

Ө1Ө2

Try this:V1 = 10 m @ 30 degrees above +x

Find: V1X = V1Y =

V2 = 10 m @ 30 degrees above –x

Find: V2X = V2Y =

Find : V3X =V3Y =

V3 = 10 m @ 30 degrees below +x

Try using the angle 330 degrees for V3

Now try this:

VX = 25m/sVY = - 51m/s

Find V=

and your point is???

• ALWAYS:

• ALWAYS:

• ALWAYS:

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

Measure counter-clockwise angles as positive

Measure clockwise angles as negative

An Example

D1

D2

D3

DT

D1 = 169 km @ 90 degrees (North)

D2 = 171 km @ 40 degrees North of East

D3 = 195 km @ 0 degrees (East)

DT = ???

D2X

D2Y

D1Y

D3X

A Review of an Example

y (km)

x (km)

But Wait. . . There’s more!

y (km)

x (km)

We’ve Found:

DTX = 326 km

DTY = 279 km.

For IDTI, use the Pythagorean Theorem.

For the Direction of DT, use Tan-1

Practice it:

• Pg. 70, # 1, 4