**VECTORS AND SCALARS**

In physics we must distinguish between **vector quantities** and **scalar quantities**.

A **vector** is an **oriented quantity**,
it has **magnitude** and **direction** like velocity, force and displacement.

**Scalars** have **no direction** associated
to them, **only magnitude**, like time, temperature, mass and energy.

**Vectors** are represented by **arrows**
where the **length of the arrow** is drawn proportionally to the **magnitude**
of the **vector**.

The letters denoting **vectors** are written
in boldface.

**1.- VECTORS ADDITION. GRAPHIC METHOD.**

To **add scalars** like mass or time, ordinary
arithmetic is used.

If two **vectors** are in the **same line**
we can also use arithmetic, but **not** if they are not in the same line.
Assume for example you walk 4 km to the East and then 3 km to the North, the
resultant or net displacement respect to the start point will have a magnitude
of 5 km and an angle
= 36.87º with the positive x direction. See figure.

The resultant displacement **V**_{R}
, is the sum of vectors **V**_{1} and **V**_{2}, that
is we write

**V**_{R} = **V**_{1}
+ **V**_{2} This is a vector equation.

The general rule to **sum vectors** in a
**graphic** way (**geometrically**) which is in fact the **definition**
how vectors are added, is the following:

(1) Use a same scale for the magnitudes.

(2) Draw one of the vectors, say **V**_{1}.

(3) Draw the other vector **V**_{2}, placing its tail on the head
of the first one, making sure to keep its direction.

(4) The sum or resultant of the vectors is the arrow drawn from the tail of
the first vector to the head of the second vector.

This method is called **vector addition**
from **tail to head**.

Notice that **V**_{1} + **V**_{2}
= **V**_{2} + **V**_{1}, that is, the order does not
matter.

This **tail to head method** can be extended
to three or more vectors. Suppose we want to add the vectors **V**_{1},
**V**_{2} and **V**_{3} shown below:

**V**_{R} = **V**_{1}
+ **V**_{3} +**V**_{3} is the resultant vector outlined
with a heavy line.

A second method to **add two vectors** is
the **parallelogram rule** equivalent to the tail to head method. In using
this parallelogram rule the two vectors are drawn from a common origin and
a parallelogram is formed using the two vectors as adjacent sides. The resultant
is the diagonal drawn from the common origin.

**2.- SUBTRACTION OF VECTORS**

Given a vector **V** it is defined the negative
of this vector (-**V**) as a vector with the same magnitude as **V**
but opposite direction:

The difference of two vectors **A** and
**B** is defined as per this equation:

**A** - **B** = **A**
+ (-**B**)

So we can use the addition rules to subtract vectors.

**3.- MULTIPLICATION OF A VECTOR BY A REAL
NUMBER.**

A vector **V** can be multiplied by a real
number c. This product is defined in such a way that c**V** has the same
direction as **V** and magnitude cV. If c is positive, the sense
is not altered. If c is negative, the sense is exactly opposite to **V**.

**More on This Theme:**

**
· Vectors, Scalars - Sum: Analytic Method
· Vectors, Addition Tools, Problems**

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