Chapter 3 Vectors
In Physics we have parameters that can be completely described by a
number and are known as “scalars” .Temperature, and mass are such
parameters
Other physical parameters require additional information about direction and
are known as “vectors” . Examples of vectors are displacement, velocity
and acceleration.
In this chapter we learn the basic mathematical language to describe
vectors. In particular we will learn the following:
Geometric vector addition and subtraction
Resolving a vector into its components
The notion of a unit vector
Add and subtract vectors by components
Multiplication of a vector by a scalar
The scalar (dot) product of two vectors (3-1)
The vector (cross) product of two vectors
An example of a vector is the displacement vector which
describes the change in position of an object as it
moves from point A to point B. This is represented by an
arrow that points from point A to point B. The length of
the arrow is proportional to the displacement magnitude.
The direction of the arrow indicated the displacement
direction.
The three arrows from A to B, from A' to B', and from A''
to B'', have the same magnitude and direction. A vector
can be shifted without changing its value if its length and
direction are not changed.
In books vectors are written in two ways:
r
Methoda
1: (using an arrow above)
Method 2: a (using bold face print)
(3-2) The magnitude of the vector is indicated by italic print: a
Geometric vector Addition
r r r
s = a +b
r
Sketch vector a using an appropriate scale
r
Sketch vector b using the same scale
r r
Place the tail of b at the tip of a
r r
The vector s starts from the tail of a
r
and terminates at the tip of b
Vector addition is commutative
r r r r
a +b = b +a
r r
Negative − b of a given vector b
r r
−b has the same magnitude as b
but opposite direction
(3-3)
Geometric vector Subtraction
r r r
d = a −b
r r r r r
( )
We write: d = a − b = a + −b
r r
From vector b we find − b
r
( )
Then we add −b to vector a
r
We thus reduce vector subtraction to
vector addition which we know how to do
Note: We can add and subtract vectors using the method of components.
For many applications this is a more convenient method
(3-4)
A component of a vector along an axis is the projection
of the vector on this axis. For example ax is the
r
projection of a along the x-axis. The component ax
is defined by drawing straight lines from the tail
r
and tip of the vector a which are perpendicular to
the x-axis.
C
From triangle ABC the x- and y-components
r
of vector a are given by the equations:
A B ax = a cos θ , a y = a sin θ
If we know ax and a y we can determine a and θ .
From triangle ABC we have:
ay
a = ax + a y
2 2
, tan θ =
ax
(3-5)
Unit Vectors (3-6)
A unit vector is defined as vector that has magnitude
equal to 1 and points in a particular direction.
A unit vector is defined as vector that has magnitude equal to 1 and points
in a particular direction. Unit vector lack units and their sole purpose is
to point in a particular direction. The unit vectors along the x, y, and z axes
ˆ
are labeled i , ˆ, and k , respectively.
ˆ j
Unit vectors are used to express other vectors
r
For example vector a can be written as:
r
a = ax i + a y ˆ .
ˆ j
The quantities ax i and a y ˆ are called
ˆ j
r
the vector components of vector a
y
r r
r Adding Vectors by Components
b
r
a x
O
r r
We are given two vectors a = ax i + a y ˆ and b = bx i + by ˆ
ˆ j ˆ j
r
We want to calculate the vector sum r = rx i + ry ˆ
ˆ j
The components rx and ry are given by the equations:
rx = ax + bx and ry = a y + by
(3-7)
y
r
r d Subtracting Vectors by Components
b
r
a x
O
r r
ˆ + a y ˆ and b = bx i + by ˆ
We are given two vectors a = ax i j ˆ j
We want to calculate the vector difference
r r r
d = a − b = d xi + d y ˆ
ˆ j
r
The components d x and d y of d are given by the equations:
d x = ax − bx and d y = a y − by
(3-8)
Multiplying a Vector by a Scalar
r r r
Multiplication of a vector a by a scalar s r esults in a new vector b = sa
The magnitude b of the new vector is given by: b = | s | a
r r
If s > 0 vector b has the same direction as vector a
r r
If s < 0 vector b has a direction opposite to that of vector a
The Scalar Product of two Vectors
r r r r
The scalar product a ⋅ b of two vectors a and b is given by:
r r
a ⋅ b =ab cos φ The scalar product of two vectors is also
known as the "dot" product. The scalar product in terms
of vector components is given by the equation:
r r
a ⋅ b =axbx + a y by + az bz
(3-9)
The Vector Product of two Vectors
r r r r r
The vector product c = a × b of the vectors a and b
r
is a vector c
r
The magnitude of c is given by the equation:
c = ab sin φ
r
The direction of c is perpendicular to the plane P defined
r r
by the vectors a and b
r
The sense of the vector c is given by the right hand rule:
r r
a. Place the vectors a and b tail to tail
r
b. Rotate a in the plane P along the shortest angle
r
so that it coincides with b
c. Rotate the fingers of the right hand in the same direction
r
d. The thumb of the right hand gives the sense of c
The vector product of two vectors is also known as
(3-10) the "cross" product
r r r
The Vector Product c = a × b in terms of Vector Components
r r
a = a xi j ˆ j ˆ r ˆ j ˆ
ˆ + a y ˆ + az k , b = b x i + by ˆ + bz k , c = c x i + c y ˆ + cz k
ˆ
r
The vector components of vector c are given by the equations:
cx = a y bz − az by , c y = az bx − ax bz , c z = ax by − a y bx
Note: Those familiar with the use of determinants can use the expression
$
i $j k$
r r
a × b = ax a y az
bx by bz
Note: The order of the two vectors in the cross product is important
r r r r
(
b ×a = − a ×b )
(3-11)
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