MCV4U
Lessons
U7L1:
Sec. 7.1/7.2 Vectors as Forces & Velocity
When we think of force, we think of things like pulling an object, lifting something, throwing. However, these
are all examples of effort or muscular exertion. What about the attraction of the moon to the Earth or the
attraction of metal to a magnet.
Force can be defined as that which changes, or tends to change, the state of rest, or uniform motion of a body.
Force is a _________ and therefore must be described using both ___________________________. On Earth,
force is defined as the product between the mass of an object and the acceleration due to gravity (9.8 m/s2). So
a 1 Kg mass exerts a downward force of 1 Kg x 9.8 m/s2 or 9.8 Kg  m 2 . The unit of measure for force is
s
called ____________ and is abbreviated as N. Therefore a 1 Kg mass exerts a force of 9.8 N downward.
Often when looking at a problem involving forces, there are many forces acting on an object. It is useful to be
able to look at a single force called __________________ which is the sum of all the forces working on the
object. The process of finding the resultant of all the forces acting on an object is called the
_____________________. If a single force that when applied to the object prevents any further motion than the
object is said to be in ____________________. The net force on the object is the zero vector.
Eg.1: Two boys push a box to the east with forces of 30 N and 40 N respectively. What is the net force.
At third boy decides to try and stop the box from moving and exerts a force of 55 N to the west. What is
the net force.
Eg.2: Two forces act on a large object. 500 N are used to pull the object south. 800 N are used to push the
object on a bearing of 100º. What is the resultant force?
Resolving a Vector into Components
In many situations involving forces, we are interested in a process that is the__________________________.
This process is called_________________, which means taking a single force and decomposing it into two
components.
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Resolve vector f into its horizontal and vertical components.
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cos  
fx
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and sin  
f
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fy

f
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Eg.3: A car is traveling at 80 km/h on a bearing of 260º, draw a vector v to represent the car’s velocity and
write the velocity in component form.
Eg.4: A mass of 20 Kg is suspended from a ceiling by two ropes that make angles of 60º and 45º with the
ceiling. Determine the tension in each of the ropes.
Homework: P.363 # 5a , 8 , 12 , 13 , 15 , 17
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P.369 # 2 , 4 , 5 , 8
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U7L2: Sec. 7.3 The Dot Product of Two Geometric Vectors
The dot product is the __________________ of 2 vectors. The result is a______________________.
The dot product for any two vectors is defined as the product of their magnitudes multiplied by the cosine of the
angle between the two vectors when the two vectors are placed in a tail-to-tail position.
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Eg.1: If a  10 and b  5 , find a b if the angle between them is 750 , 900 and 1300.
When  is acute, _________________
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When  is 900, _________________ therefore a and b are______________________.
When  is obtuse, ________________
Properties of the Dot Product
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
Commutative Law: a  b  b  a
We can change the order in the multiplication because the quantities are just scalars in the formula.

Distributive Law:
     
a  b  c   a b  a c


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 
 
a a  a a cos 0  a

Other Properties:

Associative Property with a scalar k:
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2
 
and
a 0  0
     
 
 k a   b  a  k b   k  a b 
 
 


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Eg.2: If the vectors a  3 b and 4 a  b are perpendicular, and a  2 b , determine the angle (to the nearest
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tenth of a degree) between the nonzero vectors a and b .
Homework: P.377 # 6acf , 7bde , 8 , 9 , 11 - 15
U7L3: Sec. 7.4 The Dot Product of Algebraic Vectors
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In R2 , if a  a1 , a2  and b  b1 , b2  then_______________________.
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In R3 , if a  a1 , a2 , a3  and b  b1 , b2 , b3  then________________________. Proof in textbook p.379
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Eg.1: If a  3 , 2 and b  4 ,  1 then;
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a)
Determine a b
b)
Find the angle between a and b .
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Eg.2: Determine the value of “k” such that u and v are perpendicular given u  1, 3 and v   2 , k  .
Eg.3: Evaluate the following;
 
a)
i i
 
 
b)
j j
c)
k k
 
 
d)
i j
ik
e)

 
j k
f)
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Eg.4: Find a vector (or vectors) perpendicular to each of the vectors a  0 , 2 ,1 and b  1,  1, 3
Homework: P.385 # 2 , 3 , 6bc , 7a , 9b , 10a , 11 - 14 , 17**
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U7L4: Sec. 7.5 Scalar and Vector Projections
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When two vectors, a  OA and b  OB , are placed tail to tail, and  is the angle between the vectors,
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0    180 0 , the scalar projection of a on b is ON. You can think of a projection like a shadow with the light
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shinning down from above. Ie: the shadow that a casts on b .
If the angle is____________, then the scalar projection is ______________ and if the angle is __________ than
the scalar projection is ______________. The scalar projection of vector _____________ is not equal to the
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scalar projection of_____________. Another observation is that the scalar projection of vector a on b is
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independent of the length of b .
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Calculating the scalar projection of a on b :
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We need to solve for a cos  using the dot product formula.
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Since a  b  a b cos  then
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a cos  
 
a b

b

. Similarly, b cos  
 
a b

.
a

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Eg.1: Determine the scalar projection of vector d on c if d  2 ,  3 and c  1, 4 .
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The calculation of the vector projection of a on b is the corresponding scalar projection of a on b multiplied
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by
b
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which is a unit vector pointing in the direction of b .
b
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Eg.2: Determine the vector projection of vector d on c if d  2 ,  3 and c  1, 4 .
Directional Cosines:
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Eg.3: The point R ( -5 , 3 , -3 ) is the corner of a box with another corner sitting at the origin in R3.
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a) Determine the OR .
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b) Determine the directional cosines for OR .
Homework: p. 398 # 1 , 6 , 7c , 11 , 13a , 15ab , 16** , 17**
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U7L5: Sec. 7.6 The Cross Product of Two Vectors
When we determined the dot product of two vectors the result was a___________________. The cross product
of two vectors is a __________________.
The Magnitude of the Cross Product:
To find the direction of the cross product we use the Right-Hand Rule.
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
Direct the fingers of your right hand in the direction of a .

Curl your fingers towards b .
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Your thumb will point in the direction of a b .
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The result of a b is a new vector that is perpendicular to the plane in which the given vectors lie.
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Eg.1: Determine a b then b  a for the following situations;
a)
b)
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b 4
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a 2
75º
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b 5
100º
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a 3
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 
 
 
n.b. a b  b a but the directions will be opposite  a b   b a 


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The Cross Product for Cartesian Vectors
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For a  a1 , a2 , a3  and b  b1 , b2 , b3  then ______________________________________
Trick: Write the components of the vectors in 2 rows starting with the 2nd component and repeating it at the
end. Take the downward product minus the upward product.
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Eg.2: Find a b given a  2 ,  1, 2 and b  3 , 1,  1.
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For collinear vectors, a b  a b sin  , where   0 0  a b  0
  

 
  
Combining the dot and cross products: u  v  w implies u   v  w  because  u  v   w would not be defined




since the result would be a scalar crossed with a vector. We can use this property to test if 3 vectors are
coplanar.
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 
If u , v and w are coplanar, then  v  w  is perpendicular to both v and w and since u , v and w are in the


same plane, ___________________.
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Eg.3: Determine if the following vectors are coplanar. u  0 , 8 ,  3 , v  2 ,  4 , 1 and w  1, 10 ,  3
Homework: p.408 # 4bcef , 5 , 8 , 9 , 12 , 13
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U7L6: Sec. 7.7 Applications of the Dot Product and Cross Product
When a force is acting on an object so that the object is moved from one point to another, we say that the force
has ____________________.
Eg.1: A rickshaw driver pulls his cart at an angle of 30º with the horizontal. If his current customer requires
him to pull up on the rickshaw handles with a force of 60 N, and the trip is a total of 750 m, how much
work, in J, has the rickshaw driver done over the course of this trip?
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The cross product of two vectors, defined by a and b can be used to find the area of a parallelogram.
N.B. A triangle is half of a parallelogram along the diagonals.
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Eg.2: Determine the area of QRS with vertices Q ( 2 , 4 , 1 ) , R ( 3 , -3 , 5 ) and S ( -2 , 0 , 2 ).
The cross product can also be used in the consideration of forces that involve rotation, or turning about a point

or an axis. The rotational or turning effect of a force is called __________________. F is the force applied to
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the lever (or wrench). r is the distance from the pivot point to the point where the force has been applied. Its
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
tail is at the pivot point and its head at the point where F is applied.  is the angle between r and F .
The unit for torque is the_____________________________.
Eg.3: A force of 30 N is used to open a hinged lid on the top of a box. The force is applied 30 cm from the
hinge at an angle of 45º. Calculate the magnitude of the torque.
Homework: P.415 # 3 , 5b , 6 , 7a , 8 , 9
U7L7: Unit # 7 – Review
P.418 # 2 – 7 , 8b , 9 - 12 , 15ab , 16 , 18 , 20 , 22 - 25 , 27ab , 30 , 31ab , 33 , 34
P.422 #1c 2abc 3 4 5ab 6 7
U7L8: Unit # 7 Test
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