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Some of the most important Physics Topics include energy, motion, and force.
What are the Components of a Vector and Scalar?
Geometrical Representation Of A Vector
A vector is represented by a line segment with an arrow head. The length of the line segment (in a predetermined scale), is the magnitude of the vector and the arrowhead denotes the direction. The front end (carrying the arrow) is called the head and the rear end is called tail.
In Fig., the velocity of a particle 6 cm ᐧ s-1 towards east is represented by the line segment AB. If the scale is chosen such that CD = 0.5 in. represents 2 cm ᐧ s-1, then the length of the segment AB should be 1.5 in. So, the 1.5 in. long line segment AB with the arrowhead pointing towards east represents both the magnitude and direction (towards east) of the velocity of the particle.
The velocity vector is represented as \(\overrightarrow{A B}\), where A is the initial point and B is the terminal point.
Generally, a vector is expressed as an algebraic quantity by using a letter with an arrow head like \(\vec{a}\), \(\vec{b}\), \(\vec{c}\), etc. The magnitude or absolute value of a vector is a scalar and is always positive, It is called the modulus of the vector, it is expressed as a, b, c (without arrow) or |\(\vec{a}\)|, |\(\vec{b}\)|, |\(\vec{c}\)|. Modulus of the velocity vector \(\overrightarrow{A B}\), as shown in Fig., is therefore
|\(\vec{v}\)| = |\(\overrightarrow{A B}\)| = AB(length) = 6 cm ᐧ s-1
Some Facts About Vectors
Equal vectors: Two vectors, equal in magnitude as well as in direction, are called equal vectors.
In Fig., \(\overrightarrow{A B}\) and \(\overrightarrow{C D}\) are equal vectors as both have the same magnitude (length) and direction. If \(\overrightarrow{A B}\) represents 20 towards north, then \(\overrightarrow{C D}\) will also represent 20 towards north. If \(\overrightarrow{A B}\) = \(\vec{a}\), then \(\overrightarrow{C D}\) is also = \(\vec{a}\), i.e., \(\overrightarrow{A B}\) = \(\overrightarrow{C D}\).
Two vectors may be equal even when their initial and final points are not the same. This means that we can translate a vector keeping its magnitude and direction unchanged.
Opposite vectors: Two vectors having the same absolute value but opposite directions are called opposite vectors.
In Fig., \(\overrightarrow{F E}\) and \(\overrightarrow{A B}\) or \(\overrightarrow{C D}\) are opposite vectors. \(\overrightarrow{F E}\) represents a vector with a magnitude same as that of \(\overrightarrow{A B}\) or \(\overrightarrow{C D}\) = \(\vec{a}\) (say), but the direction of \(\overrightarrow{F E}\) is opposite to that of \(\overrightarrow{A B}\) or \(\overrightarrow{C D}\). Hence, if \(\overrightarrow{A B}\) = \(\overrightarrow{C D}\) = \(\vec{a}\), then \(\overrightarrow{F E}\) = –\(\vec{a}\). It may also be written as \(\overrightarrow{A B}\) = \(\overrightarrow{C D}\) = –\(\overrightarrow{F E}\). As magnitudes or moduli of two opposite vectors are the same, we have, |\(\vec{a}\)| = a, |-\(\vec{a}\)| = a.
Collinear vectors : vectors those are of same or different magnitudes, but are parallel or anti-parallel to one another, are known as collinear vectors. In Fig., \(\vec{d}\), \(\vec{e}\), \(\vec{f}\) are collinear vectors. Also \(\vec{x}\), \(\vec{y}\), \(\vec{z}\) acting along the same line, represent a set of collinear vectors.
Coplanar vectors: Vectors lying on the same plane are coplanar vectors. In Fig., \(\vec{a}\) and –\(\vec{a}\) and all vectors in Fig. lie on the plane of the paper and hence are coplanar.
Unit vector: A vector in the direction of a given vector with unit magnitude is called unli vector. Unit vector is often denoted by a lowercase letter with circumflex or hat’ (^). Absolute value of any vector is a scalar. This scalar, multiplied by the unit vector in that direction, gives the corresponding vector.
Example: |\(\vec{A}\)| = A and A\(\hat{n}\) = \(\vec{A}\) where \(\hat{n}\) is the unit vector in the direction of \(\vec{A}\). This means \(\frac{\vec{A}}{A}\) = \(\hat{n}\). Therefore a vector divided by its magnitude gives the unit vector in the direction of that vector. In cartesian coordinate system, unit vectors along x, y and z axes are conventionally represented as \(\hat{i}\), \(\hat{j}\) and \(\hat{k}\) respectively.
Composition of Scalars And Vectors
Scalars have only magnitudes. So addition or subtraction of scalars means the addition or subtraction of their magnitudes only. Accordingly. scalar addition follows simple algebraic rules.
Vectors have both magnitude and direction. Therefore, during addition or subtraction of vectors, their directions should be taken into account as well. Thus, they cannot be added or subtracted by using simple algebraic rules. Hence, geometrical method or analytical method of vector algebra is used for vector addition.