Dot Product
The dot product of
and v  v1 , v2 is u  v  u1v1  u2 v2
u  u1 , u2
Projection of a Vector and Vector
Components



When we want a component
of a vector along a particular
direction, it is useful to think
of it as a projection.
The projection always has
length a cos q , where a is
sin fq
the length of the vector and a cos
q is the angle between the
vector and the direction
along which you want the
component.
You should know how to
write
a vector in unit vector
a  a iˆ  a ˆj or a  a , a
notation
x
y
x
y
q
qf
a cosq
Dot Product
The dot product says
something about how
parallel two vectors are.

The dot product (scalar
product) of two vectors can

be thought of as the
B
 
A  Bprojection
 AB cos q of one onto the
 direction of the other.
( A cos q ) B
ˆ

A  i  A cos q  Ax
 
A  B  Ax Bx  Ay By  Az Bz

Components
Projection is zero

B
q

p/2
A
A( B cos q )

A
Projection of a Vector: Dot
Product
The dot product says
something about how
parallel two vectors are.

The dot product (scalar
product) of two vectors can
be thought of as the
 
A  Bprojection
 AB cos q of one onto the
 direction of the other.
ˆ

A  i  A cos q  Ax
 
A  B  Ax Bx  Ay By  Az Bz

Components

B
Projection is zero
p/2

A



 
Derivation
A
How do we show that  B  A B  A B  A B ?

x
x
y
y
z
z
Start with A  A iˆ  A ˆj  A kˆ
x
y
z

B  Bxiˆ  By ˆj  Bz kˆ
 
Then A  B  ( A iˆ  A ˆj  A kˆ)  ( B iˆ  B ˆj  B kˆ)
x
y
z
x
y
z
 Axiˆ  ( Bxiˆ  By ˆj  Bz kˆ)  Ay ˆj  ( Bxiˆ  By ˆj  Bz kˆ)  Az kˆ  ( Bxiˆ  By ˆj  Bz kˆ)


But
So
iˆ  ˆj  0; iˆ  kˆ  0; ˆj  kˆ  0
iˆ  iˆ  1; ˆj  ˆj  1; kˆ  kˆ  1
 
A  B  Axiˆ  Bxiˆ  Ay ˆj  By ˆj  Az kˆ  Bz kˆ
 Ax Bx  Ay By  Az Bz
Vector operations (cont.)
The dot product (scalar product):
 The result of the dot product is a scalar:
A·B = |A||B| cosq = AB cosq
 The dot product is commutative:
B
A·B = B·A
 The dot product is distributive:
A· (B+C)=A·B+A·C
A
A·B = maximum for q = 0
A·B = minimum for q = p
A·B = zero for AB
Vector Components (cont.)
 To calculate the dot product
of two vectors, multiply like
components and add:
A·B = AxBx + AyBy + AzBz
 To calculate the cross product
of two vectors, evaluate the
following determinant:
xˆ
yˆ
zˆ
A  B  Ax
Bx
Ay
By
Az
Bz
7.3 Vectors
Definitions
Quantities such as length area, volume, temperature, and time have
magnitude only and are completely characterized by a single real number
with appropriate units (such as feet, degrees, or hours).
Such quantities are called scalar quantities, and the corresponding real
numbers are scalars.

Quantities that involve both a magnitude and a direction, such as velocity,
acceleration, and force, are vector quantities, and they can be
represented by directed line segments.

These directed line segments are called vectors.

The length of the vector represents the magnitude of the vector
quantity.

The direction is indicated by the position of the vector and the
arrowhead at one end.
We use the notation AB to name a line segment with endpoints A and B and
AB
name aAB
rayiswith
pointa Avector
and passing
through
B. A and

Thetonotation
usedinitial
to name
with initial
point
terminal point B. The vector AB terminates at B, while the ray AB
goes beyond B.

Two vectors are equal if they have the same magnitude and the same 8
7.3 Vectors
Definition: Scalar Multiplication
For any scalar k and vector A, kA is a vector with magnitude | k |
times the magnitude of A.
•
If k > 0, then the direction of kA is the same as the direction of
A.
•
If k < 0, then the direction of kA is opposite to the direction of
A.
•
If k = 0, then kA = 0 .
9
7.3 Vectors
A+B
B
A
Two forces are represented by vectors A and B.




If A and B have the same direction, then there would be a
total force A + B.
One force A + B acting along the diagonal of the
parallelogram with magnitude equal to the length of the
diagonal, has the same effect as the two forces A and B.
In physics, this result is known as the parallelogram
law.
The single force A + B acting along the diagonal is
called the sum or resultant of A and B.
10
7.3 Vectors
Note that the vector A + B coincides with the diagonal of a
parallelogram whose adjacent sides are A and B.
 A + B =B + A, and vector addition is commutative.
If A and B have the same direction or opposite direction, then no
parallelogram is formed.
 Each vector in the sum A + B is called a component of the
sum.
 For every A there is a vector – A, the opposite of A, having the
same magnitude of A but in the opposite direction.
 The sum of a vector and its opposite is the zero vector, A + (–
A) = 0.
 For any two vectors A and B, A – B = A + (– B).
Remember that in any parallelogram the opposite sides are equal
and parallel, and adjacent angles are supplementary.
The diagonals of a parallelogram do not bisect the angles of a
parallelogram unless the adjacent sides of the parallelogram are 11
7.3 Vectors
Horizontal and Vertical Components
Any nonzero vector w is the sum of a vertical component and a
horizontal component.

The horizontal component is denoted wx and the vertical component
is denoted wy .

The vector w is the diagonal of the rectangle formed by the vertical
and horizontal components.

If a vector w is placed in a rectangular coordinate system so that its
initial point is the origin, then w is called a position vector or
radius vector.

The angle q (0o q < 360o) formed by the positive x-axis and a
position vector is the direction angle for the position vector (or any
other vector that is equal to the position vector).
If the vector w has magnitude r , direction angle q, horizontal
component
w y component wy , then by using
wx wx and vertical
cos
q
and
sin q
or wx  r cos q and w y  r sin q .
trigonometric
ratios
weget

r
r
If the direction of w is such that sin q or cos q is negative, then we can
12
7.3 Vectors
Component Form of a Vector
Any vector is the resultant of its horizontal and vertical components.


Since the horizontal and vertical components of a vector determine the
vector, it is convenient to use a notation for vectors that involves them.
The notation a, b is used for the position vector with terminal point (a,
b).
a, b
Theaform
is called component0form
,0
, b . because its horizontal
component is
and its vertical component is
v  a, b
Since the vector
extends from (0, 0) to (a, b), its
magnitude is the distance between
v  a 2 these
 b 2 points:

When vectors are written in component form , operations with vectors
A  a1 , a2 and B  b1 , b2
are easier to perform.
A  B  a1  b1 , a2  b2 .
The endpoint of A + B is (a1 + b1, a2 + b2) and so
The sum can be found in component form by adding the components
instead of drawing directed line segments.
13
7.3 Vectors
Rules for Scalar Product, Vector Sum, Vector
Difference, and Dot Product
If A  a1 , a 2 , B  b1 , b2 , and k is a scalar, then

kA  ka1 , ka2

A  B  a1  b1 , a2  b2

A  B  a1  b1 , a2  b2

A  B  a1b1  a2b2
Scalar product
Vector sum
Vector difference
Dot product
14
7.3 Vectors
The Angle Between Two Vectors







If A = kB for a nonzero scalar k, then A and B are
parallel vectors.
If A and B have the same direction (k > 0) the angle
between A and B is 0o.
If they have opposite directions (k < 0) the angle
between them is 180o.
If A and B are non-parallel vectors with the same initial
point, then the vectors A, B, and A – B form a triangle.
The angle between the vectors A and B is the angle
a.
The angle between two vectors is in the interval [0o,
180o].
If the angle between A and B is 90o, then the vectors
are perpendicular or orthogonal.
15
7.3 Vectors
Theorem: Dot Product
If A and B are nonzero vectors and a is the angle between
them, then
AB
cos a 
.
AB
16
7.3 Vectors
Proof:
The vectors, A, B, and A – B form a triangle.
Apply the law of cosines to this triangle and simplify as follows.
A  B  2 A B cos a  A  B
2
2
2
a1 2  a2 2  b1 2  b2 2  2 A B cos a  a1  b1 2  a2  b2 2
 2 A B cos a  2a1b1  2a2b2
A B cos a  a1b1  a2b2
A B cos a  A  B
AB
cos a 
AB
Note that cos a = 0 if and only if A  B = 0. So two vectors are
perpendicular if and only if their dot product is zero.
Two vectors are parallel if and only cos
if a  1.
17
7.3 Vectors
y
a2 j
a1 i + a2 j
a1 i
x
Unit Vectors

The vectorsi  1, 0 and j  0, 1
are called unit
a1 , any
a2 vector
vectors because each has magnitude one. For
we have
a1 , a2  a1 1, 0  a2 0, 1  a1i  a2 j
The form a1i + a2j is called a linear combination of vectors
i and j.
 These unit vectors are thought of as fundamental vectors,
18
because any vector can be expressed as a linear
7.3 Vectors
N
v2
v3
Drift angle
W
E
v1
S





We can use the vector v1 to represent the heading and air speed of
a plane.
The vector v2 represents the wind direction and speed.
The resultant of v1 and v2 is the vector v3, where v3 represents the
course and ground speed of the plane.
The angle between the heading and the course is the drift angle.
Recall that the bearing of a vector used to describe direction in air
navigation is a non-negative angle smaller than 360o measured in a
19
clockwise direction from due north.