In physics, in addition to the units associated to the quantities,
all physical quantities are categorized in two groups:
Physics Quantities 

Scalar quantities are
defined only by their magnitude. The magnitude is the numerical value of
the quantity; example, the time of a class period is 50 minutes. The magnitude
is the number 50. The unit is the minutes. Remember that except for the
case of counting objects, units are necessary for physics quantities.
Scalar quantities are usually represented by italic type, t. 

Time, Mass 
Distance, Speed 
Vector quantities are
defined in terms of their magnitude and
direction. The magnitude of a
vector is the numerical value that defines the vector just as in the
case of scalar quantities. The direction of a vector is as important as
the magnitude when stating the characteristics of a vector. The
direction of a vector on the surface of the Earth is usually defined in
terms of the cardinal points (north, south, east, and west) or the xyplane.
Vertical directions are defined as up and down. Three dimensional
vectors are represented in the regular three dimensional coordinate
system. Vector quantities are
represented by bold type, x, and/or by
. 

Weight 
Displacement, Velocity 
In one dimension, the difference between vector quantities
and scalar quantities are minimal. The only difference is associated with the
sign of the quantity. The negative sign
provides the direction for the vector. Here, the convention used is that
vectors pointing to the right are positive while
vectors pointing to the left are negative. This
convention will be used along these notes.

Graphically, vectors can be represented by arrows
such that the lengths of the arrows are proportional to the magnitudes
of the vectors and the directions of the arrows to indicate the
direction of the vectors. The head of the arrow is also called the tip
while the bottom of the arrow is called the tail. 

Operations with Vectors 
Vectors as mathematical objects can be added, subtracted, and
multiplied by scalars as well as by vectors. Here, the previous operations will
be study from the point of view of their physical applications.

In the sequence of drawing at the left, the women originally walk
three steps toward the right; then, she advanced still in the same
direction two additional steps. At this point, she is at five steps from
her starting point which is just the addition of the 3 steps + 2 steps.
Therefore, based on observations such as the one just described, to add
vectors that are pointing in the same direction, the magnitude of the
resulting vector is obtained just from the
result of adding signed numbers, and the direction of the resulting
vector is the same as the
direction of the original vectors. Notice that this result can be
obtained following a graphical addition of
these two vectors by placing the tail of the second vector
at the tip of the first one, and the result is a vector starting at the
tail of the first vector and ending at the tip of the second vector. 

In these drawings, the woman first walk three steps
straight in the direction due East, and then four additional steps in
the direction due North. The result is that she walked from point A to
point B. Alternatively she could walk straight from point A to point B.
In this case, she will only need 5 steps in the direction straight from
A to B (see below). With respect to the eastward direction of the
drawing, the A to B direction can be calculated directly using the
trigonometric inverse function of the tangent. The definition of the
tangent is
which apply to the case of the drawing below results in

The number of steps can be calculated using the
Pythagorean theorem for right angle triangles where the three steps East
and the four steps North represent the sides of the triangle while the
hypotenuse is the straight line from A to B (the hypotenuse is the
longest side of the right angle triangle, it is also opposite to the
right angle). Thus,


Therefore, the addition of vectors that are perpendicular to
each other can be accomplish by employing the Pythagorean theorem for obtaining
the magnitude of the resultant vector and the inverse of the trigonometric
function tangent for obtaining its direction. In this case, it is also possible
to obtain the result of the vector addition using the graphical method. The
graphical method can also be used for adding vectors that are in any arbitrary
direction as demonstrated in the applet below.


On the drawing, place the first vector maintaining the magnitude
(to scale)
and direction (with respect to the positive x axis) of the original vector, (advance applet)

On the tip of the first vector, draw the second
vector maintaining the magnitude and direction (accordingly with the
horizontal axis), (advance applet)

The resultant is the vector that starts at the
tail of the first vector and end on the tip of the second vector,
(advance applet)

The characteristics of the resultant vector are
the magnitude and the direction of the vector as measured from the
horizontal line, (advance applet).

If vector R is the result of the vector
addition of vectors A and B, R = A +
B, and the magnitude of vector R is given by R =
A  B where A and B are the magnitudes
of vectors A and B respectively; then, it can be concluded
that

a) 
Vectors A and B are in the same direction. 

N 
b) 
Vectors A and B are in opposite direction. 

c) 
Vectors A and B are
perpendicular to each other 

d) 
Vectors A and B are 45^{0} from each other 

e) 
None of the above. 


Components of Vectors (Two Dimensions) 
In two dimensions, any vector can be represented as the vector addition of
two perpendicular vectors, one along the xaxis and the other along the
yaxis.
