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VECTOR ALGEBRA

SCALARS AND VECTORS

  • A quantity which has only magnitude and not related to any direction is called a scalar quantity. For example, the quantities such as mass, length, time, temperature, area, volume, speed, density, work etc. are measured by their magnitude only.
  • A quantity which has magnitude and also a direction in space is called a vector quantity. For example, the quantities such as displacement, velocity, acceleration, force, torque, etc. must be described by magnitude as well as direction.

REPRESENTATION OF VECTORS

  • Vectors are represented by directed line segments. A vector  is represented by the directed line segment PQ  such that
    • the magnitude of  vector  is equal to PQ and
    • the direction of vector  is along the line from P to Q.
  • The vector , represented by the directed line segment PQ, is denoted by  and read as 'vector PQ'. The points P and Q are respectively called the initial point and the terminal point of the vector .
  • The magnitude of the vector   is denoted by | | or simply by PQ and is equal to the distance between the points P and Q. The magnitude of a vector is also called the length or its modulus.

TYPES OF VECTORS

  1. Zero vector or Null Vector : A vector is called a zero vector if its initial and terminal points are identical. The magnitude of a zero vector is zero and the direction indeterminate. Zero vector is denoted as . For example,  is a zero vector with identical initial and terminal point P.
  2. Equal vectors : Two non-zero vectors are said to be equal vectors if their magnitudes are equal and directions are same i.e., they act parallel to each other in the same direction.
In the adjoining figure, vectors and are equal. All zero vectors are defined to be equal vectors.
  1. Collinear vectors : Two or more non-zero vectors are said to be collinear vectors if these are parallel to the same line.
Thus, non-collinear vectors are vectors which are not parallel to the same line. Hence, when their initial points coincide, they do not lie on the same line.
  1. Like and unlike vectors : Parallel Vectors are said to be like vectors if they have same direction and unlike vectors, if they have opposite directions.
The equal vectors are like vectors, but like vectors, may not be equal vectors.
  1. Unit vector : A vector is called a unit vector if its magnitude is equal to one. The sign '^' is used for a unit vector. Thus,  denotes a unit vector.
  2. Localised vector : A vector is called a localised vector if it has a specified initial point.
  3. Free vector : A vector is called a free vector if it does not have any specific initial point, and can be shifted anywhere in space keeping it parallel and direction unchanged with respect to original position. In general we assume all vectors as free vectors unless stated otherwise.
  4. Negative of a vector : The negative of a vector is defined as the vector having the same magnitude but opposite direction. For example, if , then the negative of  is the vector  and is denoted as –.
  5. Coplanar vectors : Two or more non-zero vectors are said to be coplanar vectors if these are parallel to the same plane.
Thus, non-coplanar vectors are vectors which are not parallel to the same plane. Hence, when their initial points coincide, they do not lie in the same plane.

ADDITION OF VECTORS

Let   and   be any two vectors. From the terminal point of , vector  is drawn. Then, the vector from the initial point O of to the terminal point B of  is called the sum of vectors  and and is denoted by +. This is called the triangle law of addition of vectors.
The vectors are also added by using the following method. Let  and  be any two vectors. From the initial point of , vector is drawn. Let O be their common initial point. If A and B be respectively the terminal points of   and , then parallelogram OACB is completed with OA and OB as adjacent sides. The vector is defined as the sum of   and . This is called the parallelogram law of addition of vectors.
The sum of two vectors is also called their resultant, and the process of addition as COMPOSITION.

PROPERTIES OF VECTOR ADDITION

  1. Vector addition is commutative, i.e.,
  2. Vector addition is associative,
i.e.,
  1. . So the zero vector is additive identity.
  2. . So, the additive inverse of  is .

ADDITION OF ANY NUMBER OF VECTORS

To find the sum of any number of vectors we represent the vectors by directed line segments with the terminal point of the previous vector as the initial point of the next vector. Then the line segment joining the initial point of the first vector to the terminal point of the last vector will represent the sum of the vectors.
Thus if
=

If the terminal point F of the last vector coincide with the initial point of the first vector then
i.e. the sum of vectors is a zero or null vector in this case.

DIFFERENCE OF VECTORS

If  and be any two vectors, then their difference  is defined as .

MULTIPLICATION OF A VECTOR BY A SCALAR

If  be any vector and m any scalar, then the multiplication of by m is defined as a vector having magnitude | m | | | and direction same as of , if m is positive and direction reversed, if m is negative. The product of and m is denoted by . If m = 0, then  is the zero vector.

For example,
if , then  and direction same as that of . The magnitude of the vector  –3 = 3 | | and direction opposite as that of .

PROPERTIES
  • If m and n are scalars and  any vector, then and
  • If m is any scalar and  and  any vectors, then
  • For any vector

POSITION VECTOR OF A POINT

The position vector of a point P with respect to a fixed point, say O, is the vector . The fixed point is called the origin.
Let   be any vector. We have
= Position vector of Q – Position vector of P.
 P.V. of Q – P.V. of P
A given vector can be expressed as "position vector of the terminal point minus position vector of the initial point".

COLLINEAR VECTORS

Vectors   and   are collinear iff , for some non-zero scalar .

COLLINEAR POINTS

Let A, B, C be any three points.
A, B, C are collinear   are collinear vectors.
for some non-zero scalar .

LINEAR COMBINATION OF VECTORS

Let be vectors and x, y, z, ..... be scalars then the expression is called a linear combination of vectors .
If  
Then the vector  is said to be a linear combination of vectors .

LINEARLY INDEPENDENT AND DEPENDENT SYSTEM OF VECTORS

  1. The system of vectors  is said to be linearly dependent if there exists a system of scalars x, y, z, ..... not all zero such that
  2. The system of vectors is said to be linearly independent if.

THEOREMS
  1. Two non-collinear vectorsand are linearly independent.
  2. Three non-coplanar vectorsare linearly independent.
  3. More than three vectors are always linearly dependent.

RESOLUTION OR COMPONENTS OF A VECTOR IN A PLANE

  1. Letandbe any two non-collinear vectors, then any vectorcoplanar withand, can be uniquely expressed as, where x and y are scalars.
andare called components of vectors in the directions of and respectively.
  1. Let andbe unit vectors along two perpendicular directions OX and OY respectively in a plane. These are called basic unit vectors (or Basis).
Let P(x, y) be any point in the plane
The vectors   and  are called the component vectors ofalong X-axis and Y-axis respectively.
The component vector of along X-axis is thevector
The component vector of   along Y-axis is the vector
 is the position vector of the point P(x, y)
Position vector of P(x, y) = .
Also
. This is the magnitude of
 are also called resolved parts of in the directions of  respectively.

RESOLUTION OR COMPONENTS OF A VECTOR IN SPACE

Let be three non coplanar vectors, then any vector  in space can be uniquely expressed as , where x, y, z are scalars, are the components of vectorin the directions of   respectively.
Let   be unit vectors along rectangular coordinate axes OX, OY and OZ respectively. These are called basic unit vectors, (or BASIS).
Let P (x, y, z) be any point in space. From P, draw PQ perpendicular on XY-plane. From Q, draw QA and QB perpendiculars on X-axis and Y-axis respectively.
The vectors ,   and  are called the component vectors of along X-axis, Y-axis and Z-axis respectively.
The component vectors of ,  along X-axis, Y-axis and Z-axis are respectively ,  and .  is the position vector of the point P(x, y, z).
P.V. of P(x, y, z) =
Also, OP2 = OQ2 + QP2 = (OA2 + AQ2) + QP2
= OA2 + OB2 + OC2 = | x |2 + | y |2 + | z |2 = x2 + y2 + z2
. This is the magnitude of .
,and are also called resolved parts of  in the directions of respectively.

Note that the components of a vector may be found in any given direction but the resolved parts are always in perpendicular directions.

SECTION FORMULA

Letbe the position vectors of two points A and B respectively with respect to an origin O. Then the position vectorof the point P, with respect to the same origin O, which divides AB in the ratio m : n is given by
Also,
The position vector of  the midpoint of and is . Thus, if P is the midpoint of points A and B, then .

CENTROID OF A TRIANGLE

If   be P.V.'s of the vertices A, B, C of a triangle ABC respectively, then the P.V. of the centroid G of the triangle is .
Also, the P.V. of incentre I of  is
and the P.V. of orthocentre of is .

COLLINEARITY OF THREE POINTS

The necessary and sufficient condition that three points with P.V.'sare collinear is that there exist three scalars x, y, z not all zero such that

COPLANARITY OF FOUR POINTS

The necessary and sufficient condition that four points with P.V.'s are coplanar is that there exist scalar x, y, z, t not all zero such that

SCALAR PRODUCT OF TWO VECTORS

If and  are two non-zero vectors then the scalar or dot product of   and  is denoted by. and is defined as
where  is the angle between the two vectors and.

IMPORTANT POINTS
  1. The angle between two vectors  and  is defined as the smaller angle between them when they are drawn with the same initial point.
Usually, we take . Angle between two like vectors is 0 and angle between two unlike vectors is .
  1. The dot product is a scalar quantity, since it is the product of three scalars   and .
  2. If either  is the null vector, the scalar product is the scalar zero.
  3. If   are two unit vectors, than . Thus, the scalar product of two unit vectors is equal to the cosine of the angle between their directions.
  4. Since the scalar product of two vectors is only a number, we see that (i)   is only a number being the product of two numbers   and (ii)   is a vector whose modulus is  times that of .

GEOMETRICAL INTERPRETATION OF SCALAR PRODUCT

Suppose the vectorsandare represented by the directed line segmentsandrespectively.
Then
Also,

SIGN OF THE SCALAR PRODUCT

If  be the two non-zero vectors, then the scalar product , is positive, negative, or zero, according as the angle , between the vectors is acute, obtuse, or right.

LENGTH OF A VECTOR AS A SCALAR PRODUCT

If  be any vector, then the scalar product

CONDITION OF PERPENDICULARITY

We notice that if  be any two vectors, then their scalar product  will vanish, if and only if, either at least one of the two vectors is the zero vector or the two vectors are at right angles to each other. Thus, the scalar product of two non-zero vectors is zero if they are at right angles to each other, and conversely.
being non-zero vectors.

SCALAR PRODUCT IN TERMS OF COMPONENTS

In the view of definition of scalar products of vectors, we have
Hence, if two vectors  and  are given by and , we have .

PROPERTIES OF SCALAR PRODUCTS

  1. (Commutative Law)
  2. (Distributive Law)
  3. There is no associative law for the scalar product since  is not defined as  is scalar and the scalar product is an operation on two vectors.
  4. If the scalar product of a vectorwith each of three non-coplanar vectors is zero then  must be the zero vector, because no non-zero vector can be perpendicular to three non-coplanar vectors.
In particular, if a vectoris perpendicular to every vector thenbe a zero vector.
  1. Cancellation law does not hold necessarily.
If , then  is not necessarily equal to  since is also true when  is perpendicular to . Thus   if
(i)  or (ii)  or (iii) is perpendicular to .
  1. If  and   be parallel vectors, then

SOME USEFUL VECTOR IDENTITIES

[∵ Square of a vector = square of its modulus]
  1. Similarly
  1. If  

APPLICATION OF SCALAR PRODUCT OF VECTORS

TO FIND ANGLE BETWEEN TWO VECTORS

If θ be the angle between two non-zero vectors, and   then we have  
where are the unit vectors in the directions of  and  respectively.
The angle between two non-zero vectors  and is given by .
Further, if  and . Then the angle θ between   and   is given by

TO FIND PROJECTION AND COMPONENT OF A VECTOR

We know that the scalar product of two vectors is the product of the magnitude of either vector and the projection of the other in that direction.
× Projection of   on
or × projection of  on .
It follows from the above that projection ofon =
Projection of on  =
Vector component of a vector   on
=
Similarly, the vector projection of  on .

RESOLVED PARTS
Resolved parts of a vector in the direction of a unit vector and in the direction perpendicularto  are respectively   and .
Clearly the scalar and vector components of a vector in direction of coordinates axes are respectively
and .
Hence, .

TO FIND WORK DONE BY A FORCE

Work done by a force  causing a displacement .
= displacement × component of the force along the displacement
= is the angle between and
=. = dot product of force and displacement
Let a force  displace a particle from a  point A(x1, y1, z1) to B (x2, y2, z2). Then .
=
=
Also, let
Then work done by  
=
=
[Note : The work done is zero only when  is perpendicular to.]
Suppose next that the particle is acted upon the forces    then during the displacement  of the particle the separate forces do quantities of work .
The total work done is and is therefore the same as if the system of forces were replaced by its resultant .

VECTOR OR CROSS PRODUCT OF TWO VECTORS

The vector product of two vectors and  is a vector  whose magnitude is , where   and θ is the angle between the vectors  and  and the direction ofis perpendicular to both the vectors , such that,andform a right-handed screw.
The vector product of the vectorsandis denoted byand from the above definition, we get
.....(1)
or .....(2)
where  is the angle between the vectors   and, and  is a unit vector perpendicular to both  and such that from a right-handed triad of vectors.

IMPORTANT POINTS
  1. If  or if  is parallel to , then  and
    so .
  2. We define that the vector product of any vector with a null vector is a null vector i.e. ,  if either  or .
  3. The direction of is regarded positive if the rotation from to appears to be anticlockwise.
  4. is perpendicular to the plane which contains both  and .
Thus, the unit vector perpendicular to bothandor to the plane containing  is given by .
  1. Vector product of two parallel or collinear vectors is zero. If the vectorsandare parallel or collinear, then θ = 0 or 180°, so in each case sin θ = 0. In particular.
  2. If , then  or  or andare parallel or collinear.
  3. Vector product of two perpendicular vectors
If , then , i.e, or .
  1. Vector product of two unit vectors. If and , are unit vectors, then

GEOMETRICAL INTERPRETATION OF CROSS PRODUCT

  • Let  and  be two non-parallel and non-null vectors. Let θ be the angle between OA and OB and .
    Then by definition or
.....(1)
  • Now complete the parallelogram OACB. Then area of parallelogram OACB = 2 (are Δ OAB)
=    ......From (1)
  • Hence   is a vector perpendicular to bothandwhose magnitude is equal to the area of the parallelogram whose adjacent sides are the vectorsand.
Note : Vector area of Δ OAB =
Area of Δ OAB =

PROPERTIES OF VECTOR PRODUCT

  1. Vector product is not commutative : The two vector productsandare equal in magnitude and opposite in direction.
....(i)
Hence, we conclude that .
Due to the result (i), vector product is said to be anti-commutative
  1. The vector product of a vectorwith itself is a null vector, i.e., .
  2. Vector product is associative with respect to a scalar. Thus ifandare any two vectors and m a scalar, then
  3. Distributive Law : For any three vectors,,

VECTOR PRODUCT IN TERMS OF COMPONENTS

Let be the orthonormal triad of unit vectors then we have  where  is a unit vector perp. to both  and , i.e. .   .
Similarly, . Also,
Further, as , so we have , etc.
Hence
Let
Then
=
[Using results on cross products of orthonormal triads obtained above]
i.e.

APPLICATION OF VECTOR PRODUCT

ANGLE BETWEEN TWO NON-ZERO, NON-PARALLEL VECTORS

[In terms of components]

Note : If 0 < sin θ < 1, then θ will have two value θ1 and θ2 such that 0 < θ1< 90° and 90° < θ2 < 180°. In this case angle between two vectors cannot be determined by the above formula. Hence, angle between two vectors should be evaluated using dot product.

CONDITION THAT TWO VECTORS MAY BE PARALLEL
If two vectors  and  are parallel, then , i.e.,in both cases.
Thus two vectorsare parallel if their corresponding components are proportional.

TO FIND AREA OF A TRIANGLE AND PARALLELOGRAM

  1. The vector area of a triangle ABC is equal to .
  1. The vector area of a triangle ABC with vertices having p.v.s  respectively is .
  2. The points whose p.v.'s are are collinear, if and only if
  3. represents the vector area of the parallelogram OACB whose adjacent sides areandso that form a vector triad in a right-handed system.
  1. Vector area of a quadrilateral ABCD is given by .

TO FIND VECTOR MOMENT OF A FORCE ABOUT A POINT

The vector moment or torque of a forceabout the point O is the vector whose magnitude is equal to the product of  and the perpendicular distance of the point O from the line of action of .
i.e.,
where  is the position vector of A referred to O.
The moment of a force  about O is independent of the choice of point A on the line of action of. If several forces are acting through the same point A, then the vector sum of the moments of the separate forces about a point O is equal to the moment of their resultant force about O.

TO FIND THE MOMENT OF A FORCE ABOUT A LINE

Let  be a force acting at a point A, O be any point on the given line L and  be the unit vector along the line, then moment ofabout the line L is a scalar given by .

TO FIND THE MOMENT OF A COUPLE

Two equal and unlike parallel forces whose lines of action are different are said to constitute a couple. Let P and Q be any two points on the lines of action of the forces –and  respectively.
The moment of the couple = .

TRIPLE PRODUCTS OF VECTORS

SCALAR TRIPLE PRODUCTS

For any three given vectors, the scalar product of one of the vectors and the cross product of the remaining two, is called a scalar triple product.
Thus, If  are three vectors, then   is called scalar triple product and is denoted by .

GEOMETRICAL INTERPRETATION OF SCALAR TRIPLE PRODUCT

The scalar triple product  represents the volume of a parallelepiped whose coterminous edges are represented by  which form a right handed system of vectors.
Note : The scalar triple productis positive or negative according as  form a right handed or a left handed system respectively.

EXPRESSION OF THE SCALAR TRIPLE PRODUCTIN TERMS OF COMPONENTS

Let ,
then
=
=
=
or  

PROPERTIES OF SCALAR TRIPLE PRODUCTS

  1. The scalar triple product is independent of the positions of dot and cross i.e., .
  2. The scalar triple product of three vectors is unaltered so long as the cyclic order of the vectors remains unchanged, i.e. ,
or
  1. The scalar triple product changes in sign but not in magnitude, when the cyclic order is changed. etc.
  2. The scalar triple product vanishes if any two of its vectors are equal.
i.e.
  1. The scalar triple product vanishes if any two of its vectors are parallel or collinear.
  2. For any scalar x,  
Also,
  1. For any vector
  1. Three non-zero vectors  are coplanar if and only if .
  2. Four points A, B, C, D with position vectors  respectively are coplanar if and only if , i.e., if and only if .
  3. Volume of parallelopiped with three coterminous edges
.
  1. Volume of prism on a triangular base with three coterminous edges .
  1. Volume of a tetrahedron with three coterminous edges

TO EXPRESSIN TERMS OF NON-COPLANAR VECTORS

Let  
Then ,
=
   
 

VECTOR TRIPLE PRODUCT

If  be any three vectors, then  and  are known as vector triple product
THEOREM : For any three vectors,

IMPORTANT PROPERTIES
  1. is a vector in the plane of vectors  and .
  2. , that is the cross product of vectors is not associative.
  3. , if and only if , if and only ifif if and only ifis parallel to . [  be non-zero vectors]

PRODUCTS OF FOUR VECTORS

SCALAR PRODUCT OF FOUR VECTORS

Let be four vectors. Then their scalar product is defined by .
THEOREM : For any four vectors
=

VECTOR PRODUCT OF FOUR VECTORS

Let be four vectors. Then the vector is defined as the vector product of the four vectors.
THEOREM : For any four vectors

EXPRESSING A VECTORIN TERMS OF THREE NON-COPLANAR VECTORS

Let be non-coplanar vectors, then any vector  in space can be expressed as

RECIPROCAL SYSTEM OF VECTORS

Let   form a system of non-coplanar vectors. Then the three vectors  defined by
are called reciprocal system of vectors to the system of vectors .

PROPERTIES OF RECIPROCAL SYSTEM
  1. .
  2. .
Thus, is reciprocal to the system .
  1. The orthonormal vector triad form a self-reciprocal system.
  2. If   be a system of non coplanar vectors and   be the reciprocal system of vectors, then any vector   can be expressed as

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