vectors you will be tested on your ability to: 1.correctly express a vector as a magnitude and a...
TRANSCRIPT
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