Vectors

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Vectors
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Directions can be
North
South
West
East
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or so many degrees from
30
40
S30W
E40N
50
N50E
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We add scalar quanities all the time.
Ex. 10 kg 15 kg 25 kg
However adding vectors is something new.
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The first way to add vectors is by using scale
diagrams.
Vectors can be represented graphically.
tip
tail
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Adding vectors

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Draw vectors to scale and arrange them tip to
tail.
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Draw a straight line connecting your starting
point to your end point.
This is called the resultant.
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Measure the length and the direction of the
resultant.
2 m E
4 m N
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Try this one
6 m W 4 m N
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This also works with more than two vectors.
2 m S 3 m W 4 m N
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Subtracting Vectors

• When subtracting vectors, we add the opposite.

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Another method of measuring direction is to
measure counter-clockwise from the positive
x-axis.
120
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The second way to add vectors is mathematically.
If the vectors we are adding form a right
triangle, we can use the pythagorean theorem to
solve for the resultant(hypotenuse).
c
b
a2 b2 c2
a
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Johnny walks 4.0 km N, and then turns and walks
6.0 km E. What is Johnny's total displacement?
Page 67 1,2
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However, not all vectors that we need to add
together for right triangles. In fact, if we are
adding more that 2 vectors, arranging them tip to
tail will not form a triangle at all.
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In order to add these vectors, you must be able
to find the x and y components of a vector.
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Finding vector components
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Find the x and y components.

• place on coordinate plane

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Use sin and cos to find x and y.
Page 74 11-14
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Adding Vectors
2 N
N30W

Break each vector into x and y components.
We have already done the first vector.
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Let's try the second one!
2 N
N30W
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The next step is to total all of the x and y
components
Total x x1 x2
3.1 (- 1)
2.1 N
Total y y1 y2
2.6 1.7
4.3 N
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Draw components for your x and y totals.
The hypotenuse is your result.
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To calculate the size of the result, use the
pythagorean theroem.
x2 y2 r2
(2.1)2 (4.3)2 r2
22.9 r2
4.8 N r
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To find direction, use tan
y
tan

r 4.8 N
x
y
4.3 N
4.3
2.1
x
2.1 N
tan
2.0476

64
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r 4.8 N
The final answer is
4.3 N
y
4.8 N E64N
64
x
2.1 N
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To mathematically subtract vectors
subtract the x and y components.
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A police cruiser travels 60 km S, 35 km NE,
and finally 50 kmW.
a) Calculate the total displacement.
b) If it took 1.3 hours, what was the car's
average velocity and average speed?
c) What was the car's average velocity?
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Extra practice
Glencoe Pg 67 1-4
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Relative Velocity
Suppose you are riding on a bus that is traveling
8 m/s in a positive direction. You walk at 3 m/s
towards the front of the bus.
How fast are you moving relative to the street?
If you walked 3 m/s toward the back of the bus,
what would be your velocity relative to the
street?
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Pilots use relative velocities all the time.
They must take into account the speed and
direction of the wind.
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The airspeed of a small plane is 200 km/h. The
wind speed is 50.0 km/h from the west. Determine
the velocity of the plane relative to the ground,
if the pilot keeps the plane pointing in a
direction of N40E.
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Another situation where this comes into play is
river currents.
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Ana and Sandra are riding on a ferry boat that is
traveling east at a speed of 4.0 m/s. Sandra
rolls a marble with a velocity of 0.75 m/s N,
straight across the deck of the boat to Ana.
What is the velocity of the marble relative to
the water?
Glencoe Pg 71 6-10 Pg 74 11-14 Pg 76 15-18
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Review
Page 78 11-13, 19-31
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