Vectors Review
Check Topic
Add and subtract vectors
Dot product of vectors
Find the magnitude of vectors
Find the unit vector
Notes/things to remember
Find the angle between two vectors
Find the vector projection
Decompose vectors
Determine if vectors are orthogonal or
parallel
Word problems
-Static equilibrium
-Work
-Car on slope
-Plane
-Velocity/force vector
Anything else you can think of?
Note: You should be able to apply all these to vectors in space (3D)
1. How do you add vectors?
Explain:
Practice:
v = 2i + 3j
w = 3i – 4j
v+w=
w+v=
2. How do you find the magnitude of vectors?
Explain:
Practice:
||v|| =
v = 2i + 3j
w = 3i – 4j
||w|| =
1
3. When given two positions (points), an initial and a terminal position,
how can you find the position vector?
Explain:
Practice:
Find the position vector for the given vectors:
A) Initial position P (-1, 2)
Terminal position (4, 6)
B) Initial position P (2, 3, -2)
Terminal position (3, -4, 5)
4. What is the formula to find the unit vector?
Explain:
Practice:
Find the unit vector of both v and w from #1 above.
5. What is the equation to find the velocity vector?
Explain:
Practice:
A ball is thrown with an initial speed of 25 miles per hour in a
direction that makes an angle of 30° with the positive x-axis. Write the
velocity vector v in terms of i and j. What is the initial speed in the horizontal
direction? In the vertical direction?
6. How do you find the dot product between vectors?
Explain:
Practice:
Find the dot product of v and w from #1.
Find the dot product of vectors P and Q from #3 B (the one with 3 terms!).
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7. What is the formula to find the angle between two vectors?
Explain:
Practice:
Find the angle between u = 4i – 3j and v = 2i + 5j
8. How can you tell if two vectors are parallel, orthogonal, or neither?
Explain:
Practice: Determine if the vectors below are parallel, orthogonal, or neither.
A) v = 3i – j
w = 6i – 2j
B) v = 2i – j
w = 3i + 6j
9. How do you find the vector projection of one vector onto another? How
do decompose one vector into two?
Explain:
Practice:
Find the vector projection of v = i + 3j and w = i + j.
Decompose v into two vectors v1 and v2 where v1 is parallel to w and v2 is
orthogonal to w.
Word Problem Applications
(1) WORK
W = F  AB
A girl is pulling a wagon with a force of 50 pounds. How much work is done
in moving the wagon 200 feet if the handle makes an angle of 45° with the
ground?
(2) STATIC EQUILIBRIUM
F1 + F2 + F3 = 0 rest = sum of all forces equals 0
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A weight of 800 pounds is suspended from two cables as shown in the
picture below.
What is the tension in the two cables?
(3) SPEED AND DIRECTION WITH WIND
An airplane has an airspeed of 500 km per hour bearing northeast 45°. The
wind velocity is 60 km per hour in the direction northwest 30°. Find the
resultant vector representing the path of the plane relative to the ground.
What is the ground speed of the plane? What is its direction?
(4) BRAKING LOAD OF CARS
A Toyota Sienna with a gross weight of 5300 pounds is parked on a street
with a slope of 8°. Find the force required to keep the Sienna from rolling
down the hill. What is the force perpendicular to the hill?
Try another one: A Pontiac Bonneville with a gross weight of 4500 pounds is
parked on a street with a slope of 10°. Find the force required to keep the car
from rolling down the hill. What is the force perpendicular to the hill?
MORE PRACTICE:
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1. Decompose v into 2 vectors, one parallel to w and the other orthogonal to w.
v = -3i + 2j
w = -2i + j
2. The initial point v = 3i + j – 2k. The terminal point w = -3i + 2j – k
a. Find v  w
b. Find the magnitude of || v + w||
c. Find the position vector (hint: terminal minus initial)
3. Let v = i + 4j and w = 3i – 2j
a. Find v  w
b. Find the angle between v and w
c. Find the unit vector for w
4. Let v = 2i + 3j and w = -4i – 6j
a. Find v + w
b. Find 4v – 3w
c. Find the magnitude ||v||
d. Are these vectors parallel, orthogonal, or neither? Prove it with work.
5. A weight of 2000 lbs. is suspended from 2 cables. What are the tensions in
each cable?
6. Find the work done by a force of 5 lbs acting in a direction 60° to the
horizontal in moving an object 20 feet.
7. An airplane has an airspeed of 500 km/her going in a northerly direction.
The wind velocity is 60 km/hr in a southeasterly direction. Find the actual
speed and direction of the plane relative to the ground.
8. A car with a weight of 3200 lbs is parked on a street with a slope of 12°. Find
the force required to keep the car from rolling down the hill. What is the
force perpendicular to the hill?
9. A man pushes a wheelbarrow up an incline of 20° with a force of 100 lbs.
What is the force vector in terms of i and j? What is the horizontal force? The
vertical force?
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Parametric Equations 9.7
1. Graph the parametric equation. Show its orientation.
x = 4t – 2
y=1–t
- < t < 
2. Graph the parametric equation. Show its orientation.
x = 3sin t
y = 4cos t + 2
0 ≤ t ≤ 2π
3. Find two parametric equations for the rectangular equation.
y = 2x2 – 8
4. Drew Bledsoe throws a football with an initial speed of 100 ft./sec. at an
angle of 35° to the horizontal. The ball leaves his hand at a height of 6 ft.
a. Find parametric equations that describe the position of the ball as a
function of time.
b. How long is the ball in the air?
c. When is the ball at its maximum height? What is the maximum
height?
d. Determine the distance that the ball travels.
Check Topic
Write parametric equations given a
rectangular equation
Find a rectangular equation from a
parametric equation
Graph a parametric curve
Word problem with parametric equations

Notes/things to remember
x=t
y = …… (plug in t for x)
cos2t + sin2t = 1
x 2 y 2
      1
a  b 
Chart with t, x, and y
Arrows in graph
x   v0 cos   t
1
y   gt 2   v0 sin   t  h
2
Anything else you can think of?
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