VECTOR ALGEBRA

Chapter W.R. Hamilton ( - ) Now observe that if we restrict the line l to the line segment AB, then a magnitude is prescribed on the line l with one of the two directions, so that we obtain a directed line segment (Fig . (iii)). Thus, a directed line segment has magnitude as well as direction.

Definition A quantity that has magnitude as well as direction is called a vector. Notice that a directed line segment is a vector (Fig . (iii)), denoted as or simply as , and read as ‘vector ’ or ‘vector ’. The point A from where the vector starts is called its initial point , and the point B where it ends is called its terminal point .

The distance between initial and terminal points of a vector is called the magnitude (or length) of the vector, denoted as | |, or | |, or a . The arrow indicates the direction of the vector. A Note Since the length is never negative, the notation | | < has no meaning. Position Vector From Class XI, recall the three dimensional right handed rectangular coordinate system (Fig .

(i)). Consider a point P in space, having coordinates ( x , y , z ) with respect to the origin O( , , ). Then, the vector having O and P as its initial and terminal points, respectively, is called the position vector of the point P with respect to O. Using distance formula (from Class XI), the magnitude of (or ) is given by | |= x y z In practice, the position vectors of points A, B, C, etc., with respect to the origin O are denoted by , , , etc., respectively (Fig .

(ii)). Fig . A O P ° X Y Z X A O B P( x,y,z C P( x,y,z x y z Direction Cosines Consider the position vector of a point P( x , y , z ) as in Fig . .

The angles α , β , γ made by the vector with the positive directions of x , y and z -axes respectively, are called its direction angles . The cosine values of these angles, i.e., cos α , cos β and cos γ are called direction cosines of the vector , and usually denoted by l , m and n , respectively. Fig . From Fig .

, one may note that the triangle OAP is right angled, and in it, we have . Similarly, from the right angled triangles OBP and OCP, we may write cos and cos y z β = γ = . Thus, the coordinates of the point P may also be expressed as ( lr , mr , nr ). The numbers lr , mr and nr , proportional to the direction cosines are called as direction ratios of vector , and denoted as a , b and c , respectively.

Fig . A Note One may note that l + m + n = but a + b + c ≠ , in general. . Types of Vectors Zero Vector A vector whose initial and terminal points coincide, is called a zero vector (or null vector), and denoted as .

Zero vector can not be assigned a definite direction as it has zero magnitude. Or, alternatively otherwise, it may be regarded as having any direction. The vectors represent the zero vector, Unit Vector A vector whose magnitude is unity (i.e., unit) is called a unit vector. The unit vector in the direction of a given vector is denoted by ˆ a .

Coinitial Vectors Two or more vectors having the same initial point are called coinitial vectors. Collinear Vectors Two or more vectors are said to be collinear if they are parallel to the same line, irrespective of their magnitudes and directions. Equal Vectors Two vectors are said to be equal, if they have the same magnitude and direction regardless of the positions of their initial points, and written as Negative of a Vector A vector whose magnitude is the same as that of a given vector (say, ), but direction is opposite to that of it, is called negative of the given vector. For example, vector is negative of the vector , and written as = – Remark The vectors defined above are such that any of them may be subject to its parallel displacement without changing its magnitude and direction.

Such vectors are called free vectors . Throughout this chapter, we will be dealing with free vectors only. Example Represent graphically a displacement of km, ° west of south. Solution The vector represents the required displacement (Fig .

). Example Classify the following measures as scalars and vectors. (i) seconds (ii) cm Fig . Fig .

(iii) Newton (iv) km/hr (v) g/cm (vi) m/s towards north Solution (i) Time-scalar (ii) Volume-scalar (iii) Force-vector (iv) Speed-scalar (v) Density-scalar (vi) Velocity-vector Example In Fig . , which of the vectors are: (i) Collinear (ii) Equal (iii) Coinitial Solution (i) Collinear vectors : (ii) Equal vectors : (iii) Coinitial vectors : EXERCISE . . Represent graphically a displacement of km, ° east of north.

. Classify the following measures as scalars and vectors. (i) kg (ii) meters north-west (iii) ° (iv) watt (v) – coulomb (vi) m/s . Classify the following as scalar and vector quantities.

(i) time period (ii) distance (iii) force (iv) velocity (v) work done . In Fig . (a square), identify the following vectors. (i) Coinitial (ii) Equal (iii) Collinear but not equal .

Answer the following as true or false. (i) and – are collinear. (ii) Two collinear vectors are always equal in magnitude. (iii) Two vectors having same magnitude are collinear.

(iv) Two collinear vectors having the same magnitude are equal. Fig . . Addition of Vectors A vector simply means the displacement from a point A to the point B.

Now consider a situation that a girl moves from A to B and then from B to C (Fig . ). The net displacement made by the girl from point A to the point C, is given by the vector expressed as This is known as the triangle law of vector addition. In general, if we have two vectors and (Fig .

(i)), then to add them, they are positioned so that the initial point of one coincides with the terminal point of the other (Fig . (ii)). Fig . (i) (iii) A C (ii) A C B B – –b C ’ Fig .

For example, in Fig . (ii), we have shifted vector without changing its magnitude and direction, so that it’s initial point coincides with the terminal point of . Then, the vector + , represented by the third side AC of the triangle ABC, gives us the sum (or resultant) of the vectors and i.e., in triangle ABC (Fig . (ii)), we have Now again, since , from the above equation, we have This means that when the sides of a triangle are taken in order, it leads to zero resultant as the initial and terminal points get coincided (Fig .

(iii)). Now, construct a vector so that its magnitude is same as the vector , but the direction opposite to that of it (Fig . (iii)), i.e., Then, on applying triangle law from the Fig . (iii), we have The vector is said to represent the difference of Now, consider a boat in a river going from one bank of the river to the other in a direction perpendicular to the flow of the river.

Then, it is acted upon by two velocity vectors–one is the velocity imparted to the boat by its engine and other one is the velocity of the flow of river water. Under the simultaneous influence of these two velocities, the boat in actual starts travelling with a different velocity. To have a precise idea about the effective speed and direction (i.e., the resultant velocity) of the boat, we have the following law of vector addition. If we have two vectors represented by the two adjacent sides of a parallelogram in magnitude and direction (Fig .

), then their sum is represented in magnitude and direction by the diagonal of the parallelogram through their common point. This is known as the parallelogram law of vector addition.