Introduction:

A quantity possessing both magnitude and direction, and which can be represented by a directed straight line segment, is called a vector.

Obviously, a vector related to a physical quantity will also have a unit for its magnitude. A scalar is a quantity possessing magnitude (with a unit) only. The ratio of two scalars does not have a unit.

Mathematically, a scalar is just a real number.

Examples of vectors are displacement, velocity, acceleration, force, electric field intensity, magnetic field intensity, etc.

Examples of scalars are distance, energy, voltage, permeability, etc. Finite angular rotations of a body about a point possess both magnitude and direction but they are not vectors since they cannot be represented by a directed straight line segment. Infinitesimal rotations can be treated as vectors.

__Representation of a vector:__

Pictorially, we represent a vector by a line segment having a direction. The length of the line segment is a measure of the magnitude of the vector (with/without some suitable scale) and the direction is indicated by putting an arrow anywhere on that line segment.

We write it as $\vec{AB}$ or $\vec{a}$ , and read as vector AB or a. A is called the initial point and B the terminal point (or terminus) of the vector. Line AB produced on both sides is called the line of support.

The magnitude of the vector is denoted by $ |\vec{AB}| $ or $|\vec{a}|$ (read as modulus of vector $\vec{a}$). Just letter ‘a’ can also be used to denote its magnitude.

Two vectors are said to be equal if they have (i) the same length, (ii) the same or parallel supports and (iii) the same sense.

Note:

In this chapter we will deal with only those vectors which can be moved anywhere in space protecting their magnitude and direction (called free vectors as against those which are fixed in space) i.e., the vector will be assumed unchanged if it is transferred parallel to its direction anywhere in space.

Consequently two vectors are said to be equal if they have the same magnitude and direction.

__Position vector of a point:__

We take arbitrarily any point O in space to be called the origin of reference. The position vector (p.v.) of any point P, with respect to the origin is the vector $\vec{OP}$ . For any two points P and Q in space, the equality $ \vec{PQ} = \vec{OQ} -\vec{OP} $ expresses any vector $\vec{PQ}$ in terms of the position vectors $\vec{OP}$ and $\vec{OQ}$ of P and Q respectively.

__Angle between two vectors:__

It is defined as the smaller angle formed when the initial points or the terminal points of two vectors are brought together.

Note: 0° ≤ θ ≤ 180°

Multiplication of a vector by a scalar: Given a vector $ \displaystyle \vec{a} $ and a scalar k ∈ R , then $ \displaystyle k \vec{a} $ (or $ \displaystyle \vec{a} k $ ) denotes a vector whose magnitude is $ \displaystyle |k| |\vec{a}| $

i.e., k times that of $ \displaystyle \vec{a} $ and whose direction is the same or opposite to that of $ \displaystyle \vec{a} $ according as k > 0 or k < 0 respectively.

Also, $ \displaystyle 0 \; \vec{a} = \vec{0}$ , zero or null vector which has zero magnitude and arbitrary direction.

When we have two vectors $ \displaystyle \vec{a} $ and $ \displaystyle \vec{b} $ such that $ \displaystyle \vec{b} = k \vec{a}$ , k ∈ R , then $ \displaystyle \vec{a} $ and $ \displaystyle \vec{b} $ are called collinear vectors.

$ \displaystyle \vec{b} $ is said to be a scalar multiple of $ \displaystyle \vec{a} $ .

$ \displaystyle \vec{a} $ and $ \displaystyle \vec{b} $ are parallel if k > 0 and anti parallel if k < 0.

Unit vector : A vector having the magnitude as one (unity) is called a unit vector .Unit vector in the direction of $ \displaystyle \vec{a} $ is defined as $ \displaystyle \hat{a} = \frac{\vec{a}}{|a|} $ and is denoted by $ \displaystyle \hat{a} $

__Addition of Vectors:__

Given two vectors $\displaystyle \vec{a} $ and $latex \displaystyle \vec{b} $ , their sum or resultant written as $ \displaystyle (\vec{a} + \vec{b} ) $ is a vector obtained by first bringing the initial point of $ \displaystyle \vec{b} $ to the terminal point of $ \displaystyle \vec{a} $ and then joining the initial point of $ \displaystyle \vec{a} $ to the terminal point of $ \displaystyle \vec{b} $ giving a consistent direction by completing the triangle OAB

The sum can also be obtained by bringing the initial points of $ \displaystyle \vec{a} $ and $ \displaystyle \vec{b} $ together and then completing the parallelogram OACB

Note that addition is commutative

i.e., $ \displaystyle \vec{a} + \vec{b} = \vec{b} + \vec{a} $

Also, $ \displaystyle \vec{a} + (\vec{b} + \vec{c}) = (\vec{a}+\vec{b}) + \vec{c} $

i.e. the addition of vectors obeys the associative law.

If $ \displaystyle \vec{a} $ and $ \displaystyle \vec{b} $ are collinear , their sum is still obtained in the same manner although we do not have a triangle or a parallelogram in this case.

For adding more than two vectors, we have a polygon law of addition which is just an extension of the triangle law.

$ \displaystyle \vec{OA} + \vec{AB} +\vec{BC} + \vec{CD} +\vec{DE} + \vec{EF} = \vec{OF} $

A consequence of this is that, if the terminus of the last vector coincides with the initial point of the first vector, the sum of the vectors is $ \displaystyle \vec{0} $

To obtain $ \displaystyle \vec{a} – \vec{b} $ (difference of two vectors), perform addition of a^{→} and (−b^{→})

Also, $ \displaystyle \vec{a} + \vec{0} = \vec{a}$ ;

$ \displaystyle \vec{a} + (-\vec{a})= \vec{0}$ ;

$ \displaystyle (k_1 +k_2)\vec{a} = k_1 \vec{a} + k_2 \vec{a} $ ;

$ \displaystyle k(\vec{a} + \vec{b} = k \vec{a} + k \vec{b}$ ;

Remarks:

∎ Two vectors (non-zero and non-collinear) constitute a plane. Their sum or difference also lies in the same plane. Three vectors are said to be coplanar if their line segments lie in the same plane or are parallel to the same plane.

$ \displaystyle ||\vec{a}|-|\vec{b}|| \le |\vec{a} + \vec{b}| \le |\vec{a}| + |\vec{b}|$

Illustration Show that the sum of the vectors represented by the sides AB^{→} , DC^{→}of a quadrilateral ABCD is equivalent to the sum of the vectors represented by the diagonals AC^{→} and DB^{→}

Solution:

From the triangular law of addition

$ \displaystyle \vec{AB} + \vec{BC} = \vec{AC} $

$ \displaystyle \vec{DC} + \vec{CB} = \vec{DB}$

addition of these two equation gives,

$ \displaystyle \vec{AB} + \vec{DC} = \vec{AC} + \vec{DB} $

(Since BC^{→} = – CB^{→})

Exercise :

(i) If the mid-points of consecutive sides of a quadrilateral are connected by straight lines prove that the resulting quadrilateral is a parallelogram.

(ii) ABCD is a quadrilateral and G the point of intersection of the lines joining the middle points of opposite sides. If O is any point, prove that

$ \displaystyle \vec{OA} + \vec{OB} + \vec{OC} + \vec{OD} = 4 \vec{OG}$