ELEMENTARY CONCEPTS OF VECTOR ALGEBRA
In physics, some quantities possess only magnitude and some quantities possess both magnitude and direction. To understand these physical quantities, it is very important to know the properties of vectors and scalars.
It is a property which can be described only by magnitude. In physics a number of quantities can be described by scalars.
Distance, mass, temperature, speed and energy
It is a quantity which is described by both magnitude and direction. Geometrically a vector is a directed line segment which is shown in Figure 2.10. In physics certain quantities can be described only by vectors.
Force, velocity, displacement, position vector, acceleration, linear momentum and angular momentum
The length of a vector is called magnitude of the vector. It is always a positive quantity. Sometimes the magnitude of a vector is also called ‘norm’ of the vector. For a vector , the magnitude or norm is denoted by || or simply ‘A’ (Figure 2.11).
1. Equal vectors: Two vectors and are said to be equal when they have equal magnitude and same direction and represent the same physical quantity (Figure 2.12.).
a. Collinear vectors: Collinear vectors are those which act along the same line. The angle between them can be 0° or 180°.
i. Parallel Vectors: If two vectors and act in the same direction along the same line or on parallel line, then the angle between them is 00 (Figure 2.13).
ii. Anti-parallel vectors: Two vectors and are said to be anti-parallel when they are in opposite directions along the same line or on parallel lines. Then the angle between them is 180o (Figure 2.14).
2. Unit vector: A vector divided by its magnitude is a unit vector. The unit vector for is denoted by Aˆ (read as A cap or A hat). It has a magnitude equal to unity or one.
Thus, we can say that the unit vector specifies only the direction of the vector quantity.
3. Orthogonal unit vectors: Let iˆ , jˆ and kˆ be three unit vectors which specify the directions along positive x-axis, positive y-axis and positive z-axis respectively. These three unit vectors are directed perpendicular to each other, the angle between any two of them is 90°. iˆ , jˆ and kˆ are examples of orthogonal vectors. Two vectors which are perpendicular to each other are called orthogonal vectors as is shown in the Figure 2.15
Since vectors have both magnitude and direction they cannot be added by the method of ordinary algebra. Thus, vectors can be added geometrically or analytically using certain rules called ‘vector algebra’. In order to find the sum (resultant) of two vectors, which are inclined to each other, we use (i) Triangular law of addition method or (ii) Parallelogram law of vectors.
Let us consider two vectors and as shown in Figure 2.16.
To find the resultant of the two vectors we apply the triangular law of addition as follows:
Represent the vectors and by the two adjacent sides of a triangle taken in the same order. Then the resultant is given by the third side of the triangle as shown in Figure 2.17.
To explain further, the head of the first vector is connected to the tail of the second vector . Let θ be the angle between and . Then is the resultant vector connecting the tail of the first vector to the head of the second vector . The magnitude of (resultant) is given geometrically by the length of (OQ) and the direction of the resultant vector is the angle between and . Thus we write = .
The magnitude and angle of the resultant vector are determined as follows.
From Figure 2.18, consider the triangle ABN, which is obtained by extending the side OA to ON. ABN is a right angled triangle.
From Figure 2.18
For ∆OBN, we have OB2 = ON 2 + BN 2
which is the magnitude of the resultant of and
If θ is the angle between and , then
Since vectors have both magnitude and direction two vectors cannot be subtracted from each other by the method of ordinary algebra. Thus, this subtraction can be done either geometrically or analytically. We shall now discuss subtraction of two vectors geometrically using the Figure 2.19
For two non-zero vectors and which are inclined to each other at an angle θ, the difference − is obtained as follows. First obtain − as in Figure 2.19. The angle between and − is 180-θ.