Quick
vector – usually represented by a line with an arrow that is used to represent both magnitude and direction. Commonly associated with forces, as forces have a magnitude and direction of application. For example, if one pushes something, one is pushing in a particular direction, and with a certain force. Vectors are typically represented in two or three dimensions. Often the direction is represented by an angle (or angles in three dimensions).
vector quantity – basically means a vector; it means the quantity has both a magnitude and a direction, where the direction can be represented by angles.
Nomenclature
Vectors are usually indicated by boldface type or by an arrow over the symbol. A vector quantity such as displacement is usually represented by a single letter, such as 
A |
in the figure. Vector quantities have different properties from scalar quantities; the arrow is a remider that vectors have direction.
Details
Certain quantities are described by just a number. Other quantities, in addition to a number, require a direction in order to be properly defined. For example, if you were in a car that was moving 30 mi/h to the right, and you threw a ball to the left at 20 mi/h with respect to the car, the ball would be moving at 10 mi/h with respect to a fixed point external to the car.
Some physical quantities (e.g. time, temperature, mass, density, and electric charge) can be described completely by a single number and an associated unit. But, many other quantities need a characteristic called direction to be associated with the quantity in order for it to be completely described — they can not be fully described by a number alone. Such quantities play an essential role in many of the central topics of engineering (e.g. motion and its causes, the phenomena of electricity and magnetism, forces). A simple example of a quantity with direction is the motion of an airplane. To describe this motion completely, it must be known how fast the plane is moving and in what direction. The speed of the airplane combined with its direction of motion together constitute a quantity called velocity. Another example is force. Giving a complete descripion of a force means describing both how hard the force pushes or pulls on the body and the direction of the push or pull.
Calculations with scalar quantities use the operations of ordinary arithmetic. Vector quantities require a different set of operations. To understand more about vectors and how they combine, a good starting point may be with the simplest vector quantity, displacement. Displacement is simply a change in position of a point. In the figure, the change of position from point P1 to point P2 is represented by a line from P1 to P2. An arrowhead is at P2 to represent the direction of motion. Displacement is a vector quantity because the magnitude (length) and direction are needed in order to fully be described. Walking 1 km to the north is different than walking 1 km to the west, in terms of location.
When drawing any vector, draw a line with an arrowhead at the destination end. The length of the line shows the vector's magnitude, and the direction of the line shows the vector's direction. Displacement is always a straight-line segment, directed from the starting point to the end point, even though the actual path of the particle may be curved. In the figure, the particle moves along the curved path shown from P1 to P2, but the displacement is still the vector
A |
. Note that displacement is not related directly to the total distance traveled. If the particle were to continue on to P3 and then return to P1, the displacement for the entire trip would be zero.
If two vectors have the same direction, they are parallel. If they have the same magnitude and the same direction, they are equal, no matter where they are located in space. The vector
A |
' from point P3 to point P4 in the figure has the same length and direction as the vector
A |
from P1 to P2. These two displacements are equal, even though they start at different points. This is written as
A |
=
A |
' in the figure, using a boldface equals sign to emphasize that equality of two vector quantities is not the same relationship as equality of two scalar quantities. Two vector quantities are equal only when they have the same magnitude and the same direction.
The vector
B |
in the figure, however, is not equal to
A |
because its direction is opposite to that of
A |
. The negative of a vector is defined as a vector having the same magnitude as the original vector but the opposite direction. The negative of vector quantity
A |
is denoted as −
A |
, and a boldface minus sign to emphasize the nature of the quantities. If
A |
is 87 m south, then −
A |
is 87 m north. Thus the relation between
A |
and
B |
of the figure may be written as
A |
= −
B |
or
B |
= −
A |
. When two vectors
A |
and
B |
have opposite directions, whether their magnitudes are the same or not, they are antiparallel.
The magnitude of a vector quantity (its length in the case of a displacement vector) is usually represented by the same letter used for the vector, but in light italic type with no arrow on top, rather than boldface italic with an arrow (which is reserved for vectors). An alternative notation is the vector symbol with vertical bars on both sides:
(the magnitude of
A |
) = A = |
A |
|
By definition the magnitude of a vector quantity is a scalar quantity (a number) and is always positive. It is noted that a vector can never be equal to a scalar because they are different kinds of quantities. The expression "
A |
= 6 m" is just as wrong as "2 oranges = 3 apples" or "6 lb = 7 km".
When drawing diagrams with vectors, a scale similar to those in maps are used, in which the distance on the diagram is proportional to the magnitude of the vector. For example, a displacement of 5 km might be represented in a diagram by a vector 1 cm long, since an actual-size diagram wouldn't be practical. When working with vector quantities with units other than displacement, such as force or velocity, we must use a scale. In a diagram for force vectors we might use a scale in which a vector that is 1 cm long represents a force of magnitude 5 N. A 20-N force would then be represented by a vector 4 cm long, with the appropriate direction.
Related
▪ L - Vector Product
