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# Scalars and Vectors ## Learning Objectives * Vector addition: triangle's law, parallelogram's law * Vector products: dot product and cross product * Relative velocity ## 2(b).1. Scalar and Vector Quantities Different physical quantities can be classified into the following two categories: * *...

# Scalars and Vectors ## Learning Objectives * Vector addition: triangle's law, parallelogram's law * Vector products: dot product and cross product * Relative velocity ## 2(b).1. Scalar and Vector Quantities Different physical quantities can be classified into the following two categories: * **Scalar Quantities:** Scalar quantities are those quantities which require only the magnitude for their complete specification. Examples. Mass, length, volume, density, energy, temperature, electric charge, electric potential etc. * **Vector Quantities:** Vector quantities are the quantities which require magnitude as well as direction for their complete specification. They must obey laws of vector addition. Examples. Displacement, velocity, acceleration, force, electric intensity, magnetic intensity, magnetic moment etc. **Important Notes** Electric current possesses magnitude and direction (through the conductor), but it is not a vector quantity. Electric current is a scalar quantity. ## 2(b).2. Representation of a Vector **Steps:** 1. Draw a line parallel to the direction of the vector. 2. Cut a length of the line so that it represents the magnitude of the vector on a certain scale. 3. Put an arrowhead in the direction of the vector. This line, with the arrowhead, represents the vector. A vector is written with an arrowhead on top or with the symbol "X". **Illustration:** Consider a body moving with a velocity of 40 ms-1 due east. We want to represent it vectorially. Let us select a scale, that 10 ms-1 = 1 cm. A line 4 cm long in the west-east direction from Fig. 2.b.1. with an arrowhead pointing east, represents a velocity of 40 ms-1 due east. ## 2(b).3. Some Terms Connected with Vectors * **Null Vector:** It is a vector having zero magnitude and an arbitrary direction. * **Equal Vectors:** Two vectors are equal if they possess the same magnitude and direction.They can be represented by two parallel straight lines of equal length and pointing in the same direction. * **Negative Vector:** The negative vector of another vector is the line having the same length but directed in the opposite direction. * **Co-initial Vectors:** A number of Vectors having common initial points are called co-initial vectors. * **Collinear Vectors:** Vectors having a common line of action are called collinear vectors. There are three types of collinear vectors. ## Properties of a Null Vector Following properties can be said about a null vector: * It has zero magnitude. * It has arbitrary direction. * It is represented by a point. * When a null vector is added to a given vector, the resultant vector is the same as the given vector. * Dot product of a null vector with any vector is always zero. * Cross product of a null vector with any vector is also a null vector.