Vector Analysis

Some physical quantities, such as time, temperature, mass, density and electric charge, can be described completely by a single number with a unit. But many other quantities have a direction associated with them and cannot be described by a single number. A simple example of a quantity with a direction is the motion of an airplane. To describe this motion completely, we must say not only how fast the plane is moving, but also in what direction. The speed of the airplane combined with its direction of motion together, constitute a quantity called velocity. Another example is force, which in physics means a push or pull exerted on a body. Giving a complete description of force means describing both how hard the force pushes or pulls on the body and the direction of the push or pull.

When a physical quantity is described by a single number, we call it a scalar quantity. In contrast, a vector quantity has both a magnitude ( the "how much" or "how big" part) and a direction in space. Calculations with scalar quantities use the operations of ordinary arithmetic.

To understand more about vectors and how they combine, we start with the simplest vector quantity, displacement. Displacement is simply a change in position of a point. ( The point may represent a particle or a small body.)In figure (a) we represent the change in position from point P1 to P2, with an arrowhead at P2 to represent the direction of the motion. Displacement is a vector quantity because we must state not only how far the particle moves, but also in what direction. Walking 3 km north from your front door doesn't get you to the same places as walking 3 km southeast; these two displacements have the same magnitude but different directions.

When drawing any vector, we always draw a line with an arrowhead at its tip. 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 part of the particle may be curved. In figure (b) the particle moves along the curved part shown from P1 to P2, but the displacement is still the vector . Note that the displacement is not related directly to the total distance traveled. If the particle were to continue on P3 and then return to P1, the displacement for the entire trip would be zero.

If two vectors have the same direction, they are called 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 above has the same length and direction as the vector A from point P1 to P2. These two displacements are equal, even though they start at different points. We write this as A = A' in the figure above, using a boldface equal 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 vector A because its direction is opposite ti that of vector A, we define the negative of a vector 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 we use a boldface minus sign to emphasize the vector nature of the quantities. If the vector A is 87 m south, then -A is 87 m north. When two vectors A and B have opposite directions, whether their magnitudes are the same or not, we say that they are antiparallel.

We usually represent the magnitude of a vector quantity ( its length in the case of a displacement vector ) by the same latter 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.

By definition, the magnitude of vector quantity is a scalar quantity ( a number ) and is always positive. We also note that a vector can never be equal to a scalar because they are different kinds of quantities. The expression "A= 6m" is as wrong as "2 oranges = 3 apples" or "6 lb = 7 km"!