Chapter One
Scalars and Vectors:
Statics is the study of bodies at rest. There are two quantities of importance in this
study. They are called scalars and vectors.
Scalar: a quantity that has a magnitude only (such as temperature, mass,
etc.) and can be represented by a single number
Vector: a quantity that is defined by a magnitude and a direction (such as
force, velocity, acceleration, etc.)
The above descriptions reveal that scalars and vectors are fundamentally different
entities, just like an engine and a car. An engine is in a car but an engine is not a car.
Likewise the magnitude of a vector is a scalar quantity but a vector is more than a scalar
since it also has a direction.
Since scalars and vector are different, we must use a notation which distinguishes
between them. Here we will put an underscore under a letter to denote it as a vector (v, a,
F, M) while a plain letter will be a scalar (m, s, T). Also the magnitude of the vector F
will be denoted as the scalar F. Mathematically, magnitude is denoted by a vertical bar
on each side of a vector, thus
| F | = F,
|a|=a
Conventional mathematics uses this symbol for absolute value but here it is used to
denote magnitude.
Finally we note that vectors are represented by arrows. The length of the arrow
provides the magnitude of the vector while the direction of the arrow gives the direction
of the vector. The vector V below has the length of 2 and makes equal angles with the
x and y axis.
y
V
1
1
x
To receive full credit on homework, quizzes and test, you must use the proper
notation to differentiate between scalars and vectors.
Vector Addition
Vectors are added by the head to tail method. For the two vectors V1 and V2 below
you must first draw one vector and from the end of the first you then draw the second as
shown.
y
V1
V1
V3
V2
V2
or
V2
V3
V1
x
The vector V3 which extends from the starting point of the first vector to the ending point
of the second is the vector sum V3 = V1 + V2 = V2 + V1 find the magnitude and direction
of V3, you make either of the above triangles and use the law of sines and law of cosines.
Unit Vector
A unit vector is a vector whose magnitude (or length) is one. Special symbols are
used for unit vectors in the coordinate directions. They are
i
j
k
unit vector in x direction
unit vector in y direction
unit vector in z direction
With | i | = | j | = | k | =1. Any vector V can be written as a scalar magnitude V multiplied
by a unit vector n which is in the same direction as V.
V
n
V = Vn
|n|=1
Later in this semester we will find it quite useful to write vectors in the form of a
magnitude multiplied by a unit vector.
Rectangular Components (3 Dimensions)
Here we will consider a vector V which has vector component V x, V y, V z in the x,
y, z direction shown in the right figure below.
We can write the vector V as
V = V x + V y + V z = Vx i + Vy j + Vz k
where Vx, Vy, Vz are called scalar component of V in the coordinate directions.
As shown in the left figure below, V can be defined by the three angles θ x , θ y and θ z
which it makes with the positive coordinate axis.
z
z
V
V
y
y
θy
θz
z
z
dxy
θx
z
Vx
x
Vz
Vy
x
The scalar component are related to the angles by the formulas
Vx = Vcosθ x Vy = Vcosθ y Vz = Vcosθ z
Direction cosines l, m and n are also commonly used and they are defined as the
cosines of angles as
l = cosθ x m = cosθ y n = cosθ z
Thus
Vx = Vl
Vy = Vm Vz = Vn
The magnitude of a vector can be found from the scalar components as shown in the right
figure above. Using right triangles
d xy = Vx2 + Vy2
and V2 = d2xy + Vz2
V 2 = Vx2 + Vy2 + Vz2 or V = Vx2 +Vy2 +Vz2
If one substitutes direction cosines for Vx, Vy and Vz, one obtains
V2 = V2 (l2 +m2 +n 2 ) or l2 +m2 +n 2  1
From this equation if two of the direction cosines are known, the third can be found.