Scalars
Physical quantities which can be completely specified by a number and unit, and therefore have the magnitude only, are scalars. Some physical quantities which are scalar are mass, length, time, energy etc. These examples obey the algebraic law of addition.
Vectors
Vectors are physical quantities, which besides having both magnitude and direction also obey the law of geometrical addition. (The law of geometrical addition, i.e. the law of triangular addition and law of parallelogram are discussed later in this chapter). Some physical quantities, which are vectors, are displacement, velocity, force etc.
Representation of a Vector
Since vectors have directions, any representation of them has to include the direction.
To represent a vector we use a line with an arrow head. The length of the line represents the magnitude of vector and direction of the arrow represents the direction of the vector. Let us start with a vector quantity called displacement. In the enclosed figure the change of position from point P_{1} to P_{2} is represented graphically by the directed line segment with an arrowhead to represent direction of motion.
Vector is a Physical quantity and all physical quantities have units. Hence, the vectors also have units, they are called unit vectors.
Unit Vectors: A unit vector is a vector having a magnitude of unity. Its only purpose is to describe a direction in space. On xy coordinate system ĩ denote unit vector in positive x direction and ĵ denotes unit vector in positive y direction.
Any vector in x – y plane can be represented in terms of these unit vectors ĩ & ĵ.
Similarly any vector in a 3 dimensional x y z space can be represented in terms of unit vectors ĩ, ĵ and k where, k is the unit vector in the positive z direction, as shown in figure above.
Parallel Vectors: Two or more vectors are said to be parallel when they are parallel to the same line. In the figure below, the vectors A B and C are all parallel.
Scalars and Vectors
Equal Vectors: Two or more, vectors are equal if they have the same magnitude (length) and direction, whatever their initial points. In the figure above, the vectors A and B are equal.
Negative Vectors: Two vectors which have same magnitude (length) but their direction is opposite to each, other called the negative vectors of each other. In figure above vectors A and C or B and C are negative vectors.
Null Vectors: A vector having zero magnitude is called zero vector or ‘null vector’and is written as = O vector. The initial point and the end point of such a vector coincide so that its direction is indeterminate.
The concept of null vector is hypothetical but we introduce it only to explain some mathematical results.
Invariance of the vector: Any vector is invariant so it can be taken anywhere in the space keeping its magnitude and direction same. In other words, the vectors remain invariant under translation.
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