GRAPHICAL REPRESENTATION
Vectors can be graphically represented by directed line segments. The length is chosen,
according to some scale, to represent the magnitude of the vector, and the direction of the
directed line segment represents the direction of the vector.
EQUILIBRIUM
Equilibrium, in physics, the condition of a system when neither its state of motion nor
its internal energy state tends to change with time. A simple mechanical body is said to be
in equilibrium if it experiences neither linear acceleration nor angular acceleration; unless it is
disturbed by an outside force, it will continue in that condition indefinitely. For a single particle,
equilibrium arises if the vector sum of all forces acting upon the particle is zero. A rigid body (by
definition distinguished from a particle in having the property of extension) is considered to be in
equilibrium if, in addition to the states listed for the particle above, the vector sum of all torques
acting on the body equals zero so that its state of rotational motion remains constant. An
equilibrium is said to be stable if small, externally induced displacements from that state produce
forces that tend to oppose the displacement and return the body or particle to the equilibrium
state. Examples include a weight suspended by a spring or a brick lying on a level surface. An
equilibrium is unstable if the least departure produces forces that tend to increase the
displacement. An example is a ball bearing balanced on the edge of a razor blade.
VECTOR ADDITION
The head-to-tail method is a graphical way to add vectors, described in Figure 4 below and in the
steps following. The tail of the vector is the starting point of the vector, and the head (or tip) of a
vector is the final, pointed end of the arrow.
Step 1. Draw an arrow to represent the first vector (9 blocks to the east) using a ruler and
protractor.
Step 2. Now draw an arrow to represent the second vector (5 blocks to the north). Place the tail
of the second vector at the head of the first vector.
Step 3. If there are more than two vectors, continue this process for each vector to be added.
Note that in our example, we have only two vectors, so we have finished placing arrows tip to
tail.
Step 4. Draw an arrow from the tail of the first vector to the head of the last vector. This is
the resultant, or the sum, of the other vectors.
Step 5. To get the magnitude of the resultant, measure its length with a ruler. (Note that in most
calculations, we will use the Pythagorean theorem to determine this length.)
Step 6. To get the direction of the resultant, measure the angle it makes with the reference frame
using a protractor. (Note that in most calculations, we will use trigonometric relationships to
determine this angle.)
The graphical addition of vectors is limited in accuracy only by the precision with which the
drawings can be made and the precision of the measuring tools. It is valid for any number of
vectors.
VECTOR SUBTRACTION
Vector subtraction is a straightforward extension of vector addition. To define subtraction (say
we want to subtract B from A , written A – B , we must first define what we mean by
subtraction. The negative of a vector B is defined to be –B; that is, graphically the negative of
any vector has the same magnitude but the opposite direction, as shown in Figure 13. In other
words, B has the same length as –B, but points in the opposite direction. Essentially, we just flip
the vector so it points in the opposite direction.
The subtraction of vector B from vector A is then simply defined to be the addition of –B to A.
Note that vector subtraction is the addition of a negative vector. The order of subtraction does
not affect the results.
A – B = A + (-B)
This is analogous to the subtraction of scalars (where, for example, 5 – 2 = 5 + (–2)). Again, the
result is independent of the order in which the subtraction is made. When vectors are subtracted
graphically, the techniques outlined above are used.
VECTOR COMPONENTS
In a two-dimensional coordinate system, any vector can be broken into x -component and y
-component.
v⃗=⟨vx,vy⟩
For example, in the figure shown below, the vector v⃗ is broken into two components, vx and vy.
Let the angle between the vector and its x -component be θ.
The vector and its components form a right angled triangle as shown below.
In the above figure, the components can be quickly read. The vector in the component form
is v⃗ =⟨4,5⟩v→=⟨4,5⟩
VECTOR COMPONENTS ADDITION
In this example we will be adding the two vectors shown below using the component method.
The vectors we will be adding are displacement vectors, but the method is the same with any
other type of vectors, such as velocity, acceleration, or force vectors.
