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Introduction to Vectors - Zero Vectors, Unit Vectors, Coinitial , Collinear, Equal Vectors, Addition and Subtraction of Vectors, Scalar and Vector Multiplication


                                                                                             An Introduction to Vectors 

Here's a quick look at the topics which will be covered in this tutorial.

Introducing the Vector; Terminal Point, Position Vectors, Direction Cosines

• A quantity that has magnitude as well as direction is called a 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.
• Position vector: Consider a point P in space, having coordinates (x, y, z) with respect to the origin O (0, 0, 0). 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.
• Direction cosines : Consider the position vector (or ) of a point P(x, y, z) . The angles 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, cos and cos are called direction cosines of the vector , and usually denoted by l, m and n, respectively.

Types of Vectors: Zero Vector, Unit Vector, Co-initial Vectors, Coinitial Vectors, Collinear Vectors, Equal Vectors, Negative of a Vector

Zero Vector A vector whose initial and terminal points coincide, is called a zero vector (or null vector) . Zero vector cannot 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., 1 unit) is called a unit vector.
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 and are said to be equal, if they have the same magnitude and direction regardless of the positions of their initial points .
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.


Scalar (or dot) product of two vectors :

a . b = | a ||  b | cosθ,  where, θ is the angle between a  and  b , 0 ≤θπ


Observations Regarding Scalar Products

1. a . b is a real number.

2. Let a  and  b  be two nonzero vectors, then a . b = 0 if and only if a  and  b are perpendicular to each other. i.e.

a . b = 0 a b

3. If  θ= 0, then a . b = | a || b |

In particular, a . a = | a |2, as θ in this case is 0.

4. If  θ= π, then a . b = - | a || b |

In particular, a .(- a ) = - | a |2, as θ in this case is π.

5. In view of the Observations 2 and 3, for mutually perpendicular unit vectors i, j and k, we have

i.i= j.j= k.k=1,

i.j= j.k= k.i=0

6. The angle between two nonzero vectors a  and  b  is given by

cosθ = a . b | a || b | , θ = cos-1( a . b | a || b |)

7. The scalar product is commutative. i.e.

a . b =  b . a

Multiplication of a Vector by a Scalar 

Let a  be a given vector and λ a scalar. Then the product of the vector a   by the scalar λ, denoted as λ a  is called the multiplication of vector a   by the scalar λ. Note that, λ a  is also a vector, collinear to the vector a   . The vector λ a  has the direction same (or opposite) to that of vector a  according as the value of λ is positive (or negative). Also, the magnitude of vector λ a  is | λ | times the magnitude of the vector a  , i.e.,

| λ a   | = | λ | | a   |

Vector product : 

The vector product of two nonzero vectors a  and  b  , is denoted by a x b and defined as

a x b = | a || b | sin θ n ,

where, θ is the angle between a and b  , 0 ≤ θ ≤ π and n is a unit vector perpendicular to both a and b , such that a , b and n form a right handed system (as shown in adjoining figure). i.e., the right handed system rotated from a  to  b moves in the direction of n.

Scalar triple product

Let a ,  b and  c be three vectors. Then the scalar ( a x b ).  c is called scalar triple product of   a ,  b and  c and is denoted by [ a    b    c ]

[ a    b    c ] = ( a x b ).  c

If a ,  b and  c represent the three co-terminus edges of a parallelepiped then its volume = [ a    b    c ]

Properties of Scalar Triple Products

If a ,  b and  c are cyclically permuted, the value of the scalar triple product remains unaltered.

[ a    b    c ] = [ b    c    a ] = [ c    a    b ]

The position of dot and cross can be interchanged, provided the cyclic order of vectors remains the same

a .( b x  c ) = c .( a x b ) = ( a x b ).  c

The value of scalar triple product remains the same in magnitude, but changes the sign, if the cyclic order of a ,  b and  c is changed.

The scalar triple product of three vectors is zero if any two of the given vectors are equal.

For any three vectors a ,  b and  c and a scalar λ, we have

[λ a    b    c ] = λ[ a    b    c ]

The scalar triple product of three vectors is zero if any two of the given vectors are parallel or collinear.

Vector triple product

Let a ,  b and  c be three vectors. Then the vector ( a x b )x  c is called vector triple product of   a ,  b and  c : ( a x b )x  c = ( a . c ) b - ( b . c )

( a x b )x  c is coplanar with a  and  b and is perpendicular to c .

Also, the vector triple products are not associative.

We will also cover the addition and subtraction of vectors, scalar and vector multiplication of vectors, vector triple products.

Complete Tutorial with Examples, Problems and Solutions :

In case you're interested in learning more about Vectors, here's the full set of tutorials we have :

Vectors 1a ( Theory and Definitions: Introduction to Vectors; Vector, Scalar and Triple Products)
Introducing a vector, position vectors, direction cosines, different types of vectors, addition and subtraction of vectors. Vector and Scalar products. Scalar Triple product and Vector triple product and their properties. Components and projections of vectors.

Vectors 1b ( Solved Problem Sets: Introduction to Vectors; Vector, Scalar and Triple Products )
Solved examples and problem sets based on the above concepts. 

Vectors 2a ( Theory and Definitions: Vectors and Geometry ) Vectors and geometry. Parametric vectorial equations of lines and planes. Angles between lines and planes. Co-planar and collinear points. Cartesian equations for lines and planes in 3D. 

Vectors 2b ( Solved Problem Sets: Vectors and Geometry )

Solved examples and problem sets based on the above concepts.

Vectors 3a ( Theory and Definitions: Vector Differential and Integral Calculus ) Vector Differential Calculus. Derivative, curves, tangential vectors, vector functions, gradient, directional derivative, divergence and curl of a vector function; important formulas related to div, curl and grad. Vector Integral Calculus. Line integral, independence of path, Green's theorem, divergence theorem of Gauss, green's formulas, Stoke's theorems.

Vectors 3b ( Solved Problem Sets: Vector Differential and Integral Calculus ) - Solved examples and problem sets based on the above concepts.