After reading this chapter the student should be
able to:

1.
Distinguish
between **a** **scalar **and** a vector.**

2.
Add
vectors **geometrically** and **analytically**.

3.
Be
familiar with the **unit vector** notation.

4.
Know
how to perform **vector** (or **cross) product** and **scalar** (or **dot)
product**.

A** **__vector__ has a **magnitude**
and a **direction**. Some physical quantities that are vector quantities are
displacement, velocity, acceleration and force.

A __scalar__ on the other
hand is given as a **single value** with a sign. There are many scalar quantities in physics
such as temperature, pressure, and mass.

A graphical representation of a
vector quantity in two dimensions is shown below. Its __length__ represents
the __magnitude__ of the physical quantity and the __arrow__ indicates
the __direction__. A vector has a tail and a head.

If we __translate__ a vector __without
changing its magnitude and direction__, the vector remains the same.

These three vectors are
equivalents, i.e., they represent the same vector.

Suppose an object moves along
the three paths as shown in the figure below. These three paths have the __same
displacement vector__ as they start at end at the __same points__ A and B.

Note that the distance traveled
by the object in moving from A to B is different for the three paths. It is the
biggest for the blue path.

To __add two vectors__, say
displacement vectors, means to evaluate the __net displacement__ we can use
what is called the **graphical method**. The **net** (or **resultant**)
displacement of two displacement vectors _{}and_{} is given by the vector equation_{}. The resultant is also a vector. The procedure to add the
vectors geometrically is to bring the tail of _{}at the head of _{}keeping the
orientation of the two vectors unchanged. The vector sum _{} extends from __the tail __of vector _{}to __the head__ of
vector_{}.

Next, let us add three vectors_{}, _{}, and _{}. The graphical way to do it is to perform the sum of two
vectors say _{}+_{} and then add the third one. The vector equation is (_{}+_{})+_{}. This sum is illustrated in the following diagram.

It is to be noted that _{}. The law is associative.

The vector _{}has the same magnitude
as _{}, but points in the opposite direction.

So _{}.

This is the rule of vector subtraction.

If a vector is moved from one
side of an equation to the other, a change is sign is needed similar to rules
of algebra.

It is to be noted that only
vectors of **the same kind** can be added or subtracted. We cannot add a
displacement vector to a velocity vector!

Adding vectors geometrically
does not tell us how to do the math with vector addition. We need to learn how
to calculate vector quantities mathematically. We will
do this just in two dimensions, but it can be extended to three dimensions.

A **component** of a vector
is the __projection__ of the vector on __one of the axes__ of a *set of
cartesian coordinates* *system*
as shown in the figure below.

** a_{x}** is the component of vector

Note that _{}, graphically

The values of the components *a _{x}*
and

_{}

On the other hand, if the
components *a _{x}* and

_{}(using Pythagoras theorem) and _{}

General
rule for the **sign** of the vector components:

A **unit vector** is one that
has a magnitude of 1 and points in a particular direction. It is often
indicated by putting a hat of top of the vector symbol, for example **unit****
vector** = _{}and _{}= 1.

We will label the unit vectors in the
positive directions of the x, y , and z-axes as _{}.

A two dimensional vector _{}can be expressed in terms of unit vectors as
_{}. The quantities _{}and _{}are vectors and called the **vector components** of _{}while *a _{x}*
and

We have seen in section 3.2 how
to add vectors graphically. The resolution of a vector into its components can
be used to add and subtract vectors arithmetically. To illustrate this let us
take an example. What is the sum of the following three vectors using the
components method?

_{}

_{}

_{}

The vector sum is _{}

The components of the vector _{}are:

_{}

3.6 Vectors and the Laws of
Physics

So far
we have used a convenient set of coordinate system that aligns with the edges
of your notebook. However, we can also
choose another set of coordinates as long as we are interested in the physical
vector quantity such as displacement, acceleration, force, etc. Let us take a
graphical example.

The components
of vector _{}have are different when we change the coordinates system from
(*x,y*) to (*x,y*), however, the magnitude and direction of the
vector itself have remained the same or **invariant **under **rotation**.

_{} and _{}

The new
coodinate system (*x,y*)has been formed by by rotating the old
coordinate system (*x,y*) by an angle a.

The Physics is
not affected by the choice of the coordinates system. You should always look
for the most convenient system to use in solving your problem. For example, in
the case of the inclined plane:

It is
convenient to use the (*x,y*) coordinate system rather than using the (*x,y*)
one. _{}represents the weight of the object. The object will move in
the direction of the *x*- axis. _{}is resolved in the (*x,y*) coordinate system as seen
in te figure.

The multiplication of a vector _{}by a scalar *m* gives a new vector.

_{}

If *m* > 0, the new
vector will have the same direction as the vector _{}.

If *m *< 0, the new
vector will have apposite direction to the vector _{}.

To divide _{}by *m*, we multiply _{}by _{}.

There are two types of vector
multiplication, namely, the **scalar** or **dot product** of two vectors,
which results in a scalar, and the **vector** or **cross product** of two
vectors, which results in a vector.

The **scalar
product** of two vectors _{}and _{}, denoted _{}, is defined to be

_{}

_{}and _{}are the magnitudes of vector _{}and vector_{}, respectively.

And in particular we have_{} , since the angle between a vector and itself is 0 and
the cosine of 0 is 1.

Alternatively, we have_{} , since the angle between _{} and _{} , _{}and _{}, and _{}and _{}is 90 and the cosine of 90 is 0.

The laws for scalar products are
given in the following:

_{}

The **vector product** of two
vectors _{}and _{}, denoted _{}, produces a third vector _{}whose magnitude is

_{}

The direction of _{}is perpendicular to the plane that contains the two vectors _{}and _{}as shown in the figure.

In particular
we have_{} , since the angle between a vector and itself is 0 and
the sine of 0 is 1.

Note that the angle between _{} and _{} , _{}and _{}, and _{}and _{}is 90 and the sine of 90 is 1. Therefore we use the
following diagram to help us solve problems dealing with the cross product.

However, if the rotation in the
figure is __clockwise__ such as _{}etc