2.4 Components of Vectors

In Section 2.3 we added vectors by using a scale diagram and by using properties of right triangles. Measuring a diagram offers only very limited accuracy, and calculations with right triangles work only when the two vectors are perpendicular. So we need a simple but general method for adding vectors. This is called the method of components. To define what we mean by the components of a vector A, we begin with a rectangular (Cartesian) coordinate system of axes (Figure 1). We then draw the vector with its tail at O, the origin of the coordinate system. We can represent any vector lying in the xy-plane as the sum of a vector parallel to the x-axis and a vector parallel to the y-axis. These two vectors are labeled A_x and A_y in Fig. 1.17a; they are called the component vectors of vector of A, and their vector sum is equal to A. In symbols,

Figure 1

A = A_x + A_y

Since each component vector lies along a coordinate-axis direction, we need only a single number to describe each one. When A_x points in the positive x-direction, we define the number A_x to be equal to the magnitude of A_x. When A_x points in the negative x-direction, we define the number A_x to be equal to the negative of that magnitude (the magnitude of a vector quantity is itself never negative). We define A_y in the same way.

Figure 2

We can calculate the components of the vector A if we know its magnitude A and its direction. We’ll describe the direction of a vector by its angle relative to some reference direction. In Fig. 1.17b this reference direction is the positive x-axis, and the angle between vector A and the positive x-axis is (theta). Imagine that the vector A originally lies along the postive x-axis and that you then rotate it to its correct direction, as indicated by the arrow in Fig. 2 on the angle theta. If this rotation is from the postive x-axis toward the postive y-axis, then the angle is postive, otherwise negative.

2.3 Vector Addition And Subtraction

Suppose a particle undergoes a displacement A followed by a second displacement B. The final result is the same as if the particle had started at the same initial point and undergone a single displacement C. We call displacement C the vector sum, or resultant, of displacements A and B. We express this relationship symbolically as

C = A + B

The boldface plus sign emphasize that adding two vector quantities require a geometrical process and is not same process as adding two scalar quantities as 2 +3 =5. In vector addition we usually place the tail of second vector at the head, or tip, of first vector. If we make the displacement A and B in reverse order, with B first and A second, the result is same. Thus,

C = B + A and A + B = B + A

This shows that the order of terms in a vector sum doesn’t matter. In other words, vector addition obeys the commutative law.

If A and B are both drawn with their tails at the same point, vector C is the diagonal of a parallelogram constructed with A and B as two adjacent sides.

It’s a common error to conclude that if C = A + B, then the magnitude C should equal the magnitude A plus the magnitude B. In general, this conclusion is wrong; for the vectors shown in figure above, you can see that C < A +B. The magnitude of A + B depends on the magnitude of A and B and on the angle between A and B. Only in the special case in which A and B are parallel is the magnitude of A + B equal to the sum of the magnitudes of A and B. When the vectors are anti-parallel, the magnitude of C equals the difference of the magnitudes of A and B. Be careful about distinguishing between scalar and vector quantities, and you’ll avoid making errors about the magnitude of a vector sum. When we need to add more than two vectors, we may first find the vector sum of any two, add this vectorially to the third, and so on.

We can subtract vectors as well as add them. To see how, recall that vector -A has has the same magnitude as A but the opposite direction. We define the difference A - B of two vectors A and B to be the vector sum of A and -B:

A - B = A + (-B)

A vector quantity such as a displacement can be multiplied by a scalar quantity (an ordinary number). The displacement 2A is a displacement (vector quantity) in the same direction as the vector A but twice as long; this is the same as adding A to itself. In general, when a vector A is multiplied by a scalar c, the result cA has magnitude |c|A (the absolute value of c multiplied by the magnitude of the vector A). If c is positive, cA is in the same direction as A; if c is negative, cA is in the direction opposite to A.

2.2 Conventions for Equations And Diagrams

A vector quantity V is represented by a line segment, having direction of the vector and having an arrowhead to indicate the sense. The length of the directed line segment represent to some convenient scale the magnitude of vector |V|and is printed with light-face italic type V. For example, we may choose a scale such that an arrow one inch long represents a force of twenty pounds.

In scalar equations, and frequently on diagrams where only the magnitude of a vector is labeled , the symbol will appear light-face italic type. Boldface type is used for vector quantities whenever the directional aspect of is  a part of its mathematical representation. When writing vector equations, always be certain to preserve the mathematical distinction between vectors and scalars. In handwritten work, use a distinguishing mark for each vector quantity such as an underline, V, or an arrow over the symbol to take the place of boldface type in print.

2.1 Scalars And Vectors

We use two kinds of quantities in mechanics - scalars and vectors. Scalar quantities are those with which only a magnitude is associated. Examples of scalar quantities are time, volume, density, energy, and mass. Vector, quantities on the other hand, possess direction as well as magnitude and must obey the parallelogram law of addition (to be discussed later). Examples of vector quantities are displacement, velocity, acceleration, force, moment, and momentum. Speed is a scalar. It is the magnitude of velocity, which is vector. Thus, velocity is defined by a direction as well as speed. 

Vectors representing physical quantities can be classified as free, sliding or fixed.

A free vector is one whose action is not confined to or associated with a unique line in space. For example, if a body moves without rotation, then the movement or displacement of any point in the body may be taken as a vector. This vector describes equally well the direction and magnitude of displacement of every point in the body. Thus, we may represent the displacement of such a body by a free vector.

A sliding vector has a unique line of action in space but not a unique point of application. For example, when an external force acts on a rigid body, the force can be applied at any point along its line of action without changing its effect on the body as whole, and thus it is a sliding vector.

A fixed vector is one for which a unique point of application is specified. The action of force on a deformable or non-rigid body must be specified by a fixed vector at the point of application of the force. In this instance of the forces and deformations within the body depend on the point of application of the force, as well as on its magnitude and line of action.