Vectors (Maths)

Vector

Vectors are those quantities which are completely determined if their lengths (also called magnitude) and their directions in space are given. For example, velocity, acceleration, force etc. are vectors.


Directed line segment

A vector or a directed line segment is denoted by two letters with an arrow marked over them. The first letter is the starting point (or the initial point or the origin) and the second letter is the terminal point of a vector. Thus, is a directed line segment whose initial point is A and the terminal point is B. BA is a directed line segment whose initial point is B and the terminal point A.



The length, AB, of the line segment is known as the magnitude of vector and is written as or simply by AB. Vectors and have the same magnitude but the directions are

opposite. Hence, and represent different vectors. We write, etc. to represent vectors and, etc. to represent their magnitudes.



Free and localized vectors

If we are free to choose the origin (or the initial point) of a vector at any point, then the vector is said to be a free vector. A free vector can be subjected to parallel displacement to itself without changing its magnitude (and direction, of course). If the initial point is restricted to a specified point, then the vector is called a localised vector.


Equal vectors
Two vectors and are said to be equal, if they have the same length (magnitude) and the same direction, regardless of the position of their initial points, and we write = . Thus, equal vectors can be represented by parallel lines of equal lengths, in the same sense of direction, irrespective of the initial point.




Unit vector

A vector is said to be a unit vector if its magnitude is of unit length. The unit vector, corresponding to a vector of length and in the direction of, is. We denote it by.



Zero vector

A vector is said to be zero or a null vector if its magnitude is zero, i,e. = 0. Its direction is indeterminate. We represent zero vector by ., are zero vectors.

Distinction must be made between scalar O and zero vector . The scalar O is a real number whereas is a vector of zero magnitude and arbitrary direction.


Collinear vectors

A number of given vectors are said to be collinear if they are parallel to (or coincident with) the same line irrespective of their magnitude. Since zero vector can have any direction, it is collinear with any other vector. Collinear or parallel vectors are also called like vectors.



Negative of a vector
A vector whose magnitude is the same as that of a given vector but is in the opposite direction is called the negative of and is denoted by . The negative of a vector is collinear with the vector itself.



Coplanar vectors
A number of given vectors are said to be coplanar if they are parallel to the same plane. Obviously, vectors lying in the same plane are coplanar.



Addition of vectors
In the side wise figure the sum of the vectors and is defined as. Obviously, we can choose any point as origin. Hence, the sum of two vectors is independent of the choice of initial point (or origin). Since, OAB is a triangle, this is also called the triangle law of addition
.

Moreover, the magnitude of is not equal to the sum of the magnitude of vectors and . If , i,e. O and B coincide, then or is negative of .

Vector addition is commutative i,e. .

Vector addition is associative i,e. .


The process of addition may be extended to several vectors. Thus , to add n vectors , we choose O as the origin and draw

so that

Hence the sum is represented by . This is called the polygon law of addition.

If three vectors are represented by three sides of a triangle, taken in order, then. If vector, taken in order, are represented by then sides of a

polygon, then .

The difference of two vectors and can be defined as the sum of the vectors and i,e. .

We have, ,

.




The definitions of the sum and the difference of two vectors show that and .

Moreover ,.


Scalar multiplication of vectors

The multiplication of a vector by a scalar m is defined as the vector, whose magnitude (or modules) is times the magnitude of. This vector has the same direction as, if m is positive and has the opposite direction of if m is negative. Also,


.


(m, n scalars).





We have, and. If and are like vectors, then, so that

or, .

Hence, one vector can be expressed as a scalar multiple of the other.


Position vector of a point

Let O be the origin of a certain co-ordinate system. Then every point in space can be referenced with respect to this origin. If P is any point in space, then is said to be the position vector of the point P. Position vector is unique.



Position vectors in rectangular Cartesian system:

In the three dimensional co-ordinate (rectangular) system, unit vectors in the direction of positive x-axis,

y-axis and z-axis are denoted by respectively. In the side wise figure The .

Now

= .

Vectors are called the components of vector in the directions of the co-ordinate axes. (x, y, z) is sometimes called the position vector of P(x, y, z) and we denote it as .

Also

(OP)² =

.

if A(x₁, y₁, z₁) and B(x₂, y₂, z₂) be any two points in space. Then vector can be written as (using (1)),

where a₁ = x₂ - x₁, a₂ = y₂ - y₁, a₃ = z₂ - z₁.

We find that any vector can be expressed in term of unit vectors . These are known as basis (or base) vectors. In general, any three mutually perpendicular (or orthogonal)

vectors, say can be chosen as base vectors.

The points A, B, C whose position vectors are, are collinear if AB + BC = AC

or if

i.e. if.


Director Cosines of a Vector:

Let , , be the angles which the vector makes with the positive directions of the x-axis, the y-axis and z-axis respectively. The cosine of these angles i.e. cos, cos and cos are called the direction cosines of the vector . The direction cosines are usually denoted by l, m, n respectively. From the right angled triangles OAP, OBP and OCP, we find that cos = ,

cos = , cos =

cos² + cos² + cos² = .

The direction cosines (l, m, n) in this order, thus constitute a unit vector along the vector . For any non-zero constant k, kl, km, kn are called the direction ratios of .


Dividing a line segment in a given ratio (Section formula)

In the side wise figure the position vector of P is






If P is the mid-point of AB, then = , and hence the position vector of P is .
The position vector of the point P which divides AB (externally) in the ratio : is .

Linear Combination of vectors:

Two vectors are collinear (or parallel) if their directions are the same or opposite, whatever their magnitude.

Let us consider three vector lying in a plane. If the initial points of and are different, we shift the vectors (by parallel sliding) so that all the three have the same initial point O.

Any vector in the plane of two non-collinear vectors and can be represented as a linear combination of and .If and and are non-collinear, then m = 0, n = 0.

If are non-coplanar vectors then any vector can be represented as a linear combination of.


Product of two vectors:

The product of two numbers is a number. But the product of two vectors is defined in two ways, namely, scalar product where the result is a scalar and vector product where the result is a vector quantity.


Scalar product of two vectors (dot product):

The scalar product of two vectors is given by |a| |b| cos, where (0 ) is the angle between the vectors. It is denoted by , and for q = 0, .

, as = 0

Two vectors make an acute angle if, an obtuse angle if and are at right angle if



.

,

.

If i, j and k are three unit vectors along three mutually perpendicular lines, then

= 1 and.

If, then

and , cos = = .

If , then it does not always imply that. Since

, it is possible that is perpendicular to vector .

For a scalar ,


Components of a vector along the coordinate axes:

If is a vector which makes angles , , with the coordinate axes, then



and cos = are the corresponding direction cosines of the vector.



Vector product of two vectors (cross product):

The vector product of two vectors , written as , is the vector sinq, where q is the angle between , (0 ), and is a unit vector along the line

perpendicular to both . The direction of is such that form a right - handed system. Note that

, ,

.

.

Two non - zero vectors are collinear if and only if.



.


Also,

= 2

= 2 (Area of triangle AOC) = Area of parallelogram OABC.

The area of the parallelogram, whose adjacent sides are represented by vectors and and the area of the triangle OAB is . Here is said to be the

vector area of the parallelogram with adjacent sides OA and OB.

The unit vector perpendicular to the plane of , and a vector of magnitude l perpendicular to the plane of .

= .




Scalar triple product:

The scalar triple product of three vectors is defined as We denote it by [a b c].

i.e. [a b c]= [b c a] = [c a b] = -[b a c] = -[c b a].

If =

then = = .


represents (and is equal to) the volume of the parallelopiped whose adjacent sides are represented by the vectors .


The volume of the tetrahedron ABCD is equal to.

Three vectors are coplanar if [a b c] = 0.

Moreover, [a + b, c, d] = [a c d] + [b c d].

,


[a b c] [u v w] = .


Four points with position vectors are coplanar if

[d b c] + [d c a] + [d a b] = [a b c].






Vector triple product:

The vector triple product of three vectors is the vector:



Note:
= =

= = .

This is called Lagrange's identity.

.


Let be a system of, three non - coplanar vectors. Then the system of vectors which satisfies and is called the reciprocal system to the vectors . In term of the vectors are given by .


Geometrical applications:

Let be unit vectors along two lines inclined at an angle Then are the vectors along the internal and external bisectors of , respectively. The bisectors of the angles

between the lines are given by ( real).