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VECTORS
Pythagorean Theorem
• Recall that a right triangle has a 90° angle as one of its
angles.
• The side that is opposite the 90° angle is called the
hypotenuse.
• The theorem due to Pythagoras says that the square of
the hypotenuse is equal to the sum of the squares of the
legs.
c2 = a2 + b2
a
c
b
Trigonometry
Trigonometry is concerned with the
connection between the sides and angles in
any right angled triangle.
Angle
Introduction to Trigonometry
• In this section we define the three basic
trigonometric ratios, sine, cosine and tangent.
• opp is the side opposite angle A
• adj is the side adjacent to angle A
• hyp is the hypotenuse of the right triangle
hyp
opp
adj
A
There are three formulae involved in
trigonometry:
sin A=
cos A=
tan A =
S OH C AH T OA
Using ratios to find angles
We have just found that a scientific calculator
holds the ratio information for sine (sin),
cosine (cos) and tangent (tan) for all angles.
It can also be used in reverse, finding an
angle from a ratio.
To do this we use the sin-1, cos-1 and tan-1
function keys.
Finding an angle from a triangle
To find a missing angle from a right-angled triangle
we need to know two of the sides of the triangle.
We can then choose the appropriate ratio, sin, cos or
tan and use the calculator to identify the angle from
the decimal value of the ratio.
1.
14 cm
6 cm
θ
Find angle θ
a) Identify/label the names of
the sides.
b) Choose the ratio that
contains BOTH of the
letters.
Finding a side from a triangle
To find a missing side from a right-angled triangle we
need to know one angle and one other side.
Note: If
Cos45o
=
x
13
To leave x on its own we need to move the
÷ 13.
It becomes a “times” when it moves.
Cos45o x 13 = x
Finding a side from a triangle
There are occasions when the unknown letter is on
the bottom of the fraction after substituting.
Cos45o =
13
u
Move the u term to the other side.
It becomes a “times” when it moves.
Cos45o x u = 13
To leave u on its own, move the cos 45 to other
side, it becomes a divide.
u =
13
Cos 45
Values of Trigonometric Functions for Common Angles
θ
0o
30o
37o
45o
53o
60o
90o
sinθ
0
0.500
0.600
0.707
0.800
0.866
1.00
cosθ
1.00
0.866
0.800
0.707
0.600
0.500
0
tanθ
0
0.577
0.750
1.00
1.33
1.17
∝
Vector vs. Scalar Review
• All physical quantities encountered in this text will be
either a scalar or a vector
• A vector quantity has both magnitude (size) and direction
• A scalar is completely specified by only a magnitude
(size)
Vector Notation
• When handwritten, use an arrow:
• When printed, will be in bold print with an arrow:
A
• When dealing with just the magnitude of a vector in print,
an italic letter will be used: A
Properties of Vectors
• Equality of Two Vectors
• Two vectors are equal if they have the same magnitude and the
same direction
• Movement of vectors in a diagram
• Any vector can be moved parallel to itself without being affected
More Properties of Vectors
• Negative Vectors
• Two vectors are negative if they have the same magnitude but are
180° apart (opposite directions)
•
 
A  B; A  A  0
• Resultant Vector
• The resultant vector is the sum of a given set of vectors
•
R  A B
Adding Vectors
• When adding vectors, their directions must be taken into
account
• Units must be the same
• Geometric Methods
• Use scale drawings
• Algebraic Methods
• More convenient
Adding Vectors Geometrically (Triangle or
Polygon Method)
• Choose a scale
• Draw the first vector with the appropriate length
and in the direction specified, with respect to a
coordinate system
• Draw the next vector with the appropriate length
and in the direction specified, with respect to a
coordinate system whose origin is the end of
vector A and parallel to the coordinate system
used for A
Graphically Adding Vectors, cont.
• Continue drawing the
vectors “tip-to-tail”
• The resultant is drawn
from the origin of A to the
end of the last vector
• Measure the length of R
and its angle
• Use the scale factor to
convert length to actual
magnitude
Graphically Adding Vectors, cont.
• When you have many
vectors, just keep
repeating the process
until all are included
• The resultant is still
drawn from the origin of
the first vector to the end
of the last vector
Notes about Vector Addition
• Vectors obey the
Commutative Law
of Addition
• The order in which the
vectors are added
doesn’t affect the result
•
A B B A
Vector Subtraction
• Special case of
vector addition
• Add the negative of the
subtracted vector
• A  B  A  B
 
• Continue with
standard vector
addition procedure
Multiplying or Dividing a Vector by a
Scalar
• The result of the multiplication or division is a
vector
• The magnitude of the vector is multiplied or
divided by the scalar
• If the scalar is positive, the direction of the result
is the same as of the original vector
• If the scalar is negative, the direction of the result
is opposite that of the original vector
Components of a Vector
• A component is a
part
• It is useful to use
rectangular
components
• These are the
projections of the
vector along the x- and
y-axes
Components of a Vector, cont.
• The x-component of a vector is the projection along the x-
axis
Ax  A cos 
• The y-component of a vector is the projection along the y-
axis
Ay  A sin
• Then,
A  Ax  Ay
More About Components of a Vector
• The previous equations are valid only if θ is measured
with respect to the x-axis
• The components can be positive or negative and will have
the same units as the original vector
More About Components, cont.
• The components are the legs of the right triangle
whose hypotenuse is A
A
2
x
2
y
A A
and
 Ay 
  tan  
 Ax 
1
• May still have to find θ with respect to the positive x-axis
• The value will be correct only if the angle lies in the first
or fourth quadrant
• In the second or third quadrant, add 180°
Adding Vectors Algebraically
• Choose a coordinate system and sketch the vectors
• Find the x- and y-components of all the vectors
• Add all the x-components
• This gives Rx:
Rx   v x
Adding Vectors Algebraically, cont.
• Add all the y-components
• This gives Ry: R 
v
y

y
• Use the Pythagorean Theorem to find the magnitude of
the resultant:
R  R2x  R2y
• Use the inverse tangent function to find the direction of R:
  tan
1
Ry
Rx