A Vector may be described as a quantity having both magnitude & direction. A vector is generally represented by a directed line segment, say . A is called the initial point & B is called the terminal point. The magnitude of vector is expressed by.
Zero vector a vector of zero magnitude i.e.which has the same initial & terminal point, is called a Zero vector. It is denoted by O.
Unit vector a vector of unit magnitude in direction of a vector is called unit vector along and is denoted by aˆ symbolically
Equal vectors two vectors are said to be equal if they have the same magnitude, direction & represent the same physical quantity.
Collinear vectors two vectors are said to be collinear if their directed line segments are parallel disregards to their direction. Collinear vectors are also called Parallel vectors. If they have the same direction they are named as like vectors otherwise.
unlike vectors. Symbolically, two non zero vectors are collinear if and only if,
where K ∈ R Coplanar vectors a given number of vectors are called coplanar if their line segments are all parallel to the same plane. Note that “two vectors are always coplanar”. position vector let O be a fixed origin, then the position vector of a point P is the vector →OP . If a & b & position vectors of two point A and B, then , →AB = b a
− = pv of B − pv of A .
2. Vector addition:
If two vectors are represented by → , then their sum is a vector represented by , where OC is the diagonal of the parallelogram OACB.
3. Multiplication of vector by scalars :
If is a vector & m is a scalar, then m is a vector parallel to whose modulus is |m| times that of . This multiplication is called Scalar multiplication. If are vectors & m, n are scalars, then:
4. Section formula :
If are the position vectors of two points A & B then the p.v. of a point which divides AB in the ratio m : n is given by :
note p.v..of mid point of AB =
5. Direction Cosines
Let the angles which this vector makes with the +ve directions OX,OY & OZ are called Direction angles & their cosines are called the Direction cosines Note that, cos² αααα + cos² ββββ + cos² ΓΓΓΓ = 1.
6. Vector equation of a line:
Parametric vector equation of a line passing through two point
is given by, where t is a parameter. If the line passes through the point & is parallel to the vector then its equation is,
Note that the equations of the bisectors of the angles between the lines
7. Test of collinearity :
Three points A,B,C with position vectors
respectively are collinear, if & only if there exist scalars x , y , z not all zero simultaneously such that ; where x + y + z = 0.
8. Scalar product of two vectors:
note that if θ is acute then & if θ is obtuse then
Note: That vector component of and perpendicular to the angle φ between is given by cos φ = 0 ≤ φ ≤ π.
Note : (i) Maximum value of
(ii) Minimum values of
(iii) Any vector can be written as
(iv) A vector in the direction of the bisector of the angle between the two vectors Hence bisector of the angle between the two vector where Bisector of the exterior angle between
9. Vector product of two vectors :
(i) If are two vectors & θ is the angle between them then where is the unit vector perpendicular to both such that forms a right handed screw system.
(ii) Lagranges Identity : for any two vectors
(iii) Formulation of vector product in terms of scalar product:
The vector product is the vector such that.
(i) form a right handed system.
(iv) are parallel (collinear) where K is a scalar..
(vi) Geometrically of the parallelogram whose two adjacent sides are represented by
(vii) Unit vector perpendicular to the plane of
- A vector of magnitude ‘r ’ & perpendicular to the palne of
If θ is the angle between
(viii) Vector area If are the pv’s of 3 points A, B & C then the vector area of triangle ABC The points A, B & C are collinear if
Area of any quadrilateral whose diagonal vectors are
10. Shortest distance between two lines:
If two lines in space intersect at a point, then obviously the shortest distance between them is zero. Lines which do not intersect & are also not parallel are called SKEW LINES. For Skew lines the direction of the shortest distance would be perpendicular to both the lines. The magnitude of the shortest distance vector would be equal to that of the projection of along the direction of the line of shortest distance, is parallel to i. e .
1. The two lines directed along will intersect only if shortest distance = 0 i.e.
lies in the plane containing
2. If two lines are given by i.e. they are parallel then
11. Scalar triple product / box product / mixed product :
The scalar triple product of three vectors is defined as :
sin θ cos φ where θ is the angle between it is also defined as spelled as box product .Scalar triple product geometrically represents the volume of the parallelopiped whose three couterminous edges are represented by
In a scalar triple product the position of dot & cross can be interchanged i.e.
where are non coplanar vectors .
If are coplanar
Scalar product of three vectors, two of which are equal or parallel is 0 i.e.
Note : If are non − coplanar then for right handed system & for left handed system .
The volume of the tetrahedron OABC with O as origin & the pv’s of A, B and C being respectively is given by
The positon vector of the centroid of a tetrahedron if the pv’s of its angular vertices are are given by
Note that this is also the point of concurrency of the lines joining the vertices to the centroids of the opposite faces and is also called the centre of the tetrahedron. In case the tetrahedron is regular it is equidistant from the vertices and the four faces of the tetrahedron .
12. Vector Triple Product : Let be any three vectors, then the expression is a vector & is called a vector triple product .
Geometrical interpretation of
Consider the expression which itself is a vector, since it is a cross product of two vectors Now is a vector perpendicular to the plane containing but is a vector perpendicular to the plane therefore is a vector lies in the plane of and perpendicular to Hence we can express in terms of i.e. where x & y are scalars .
13. Linear combinations / Linearly Independence and Dependence of Vectors :
Given a finite set of vectors then the vector
is called a linear combination of for any x, y, z ...... ∈ R. We have the following results :
(a) Fundamental theorem in plane : Let be non zero , non collinear vectors . Then any vector coplanar with can be expressed uniquely as a linear combination of There exist some unique x,y ∈ R such that
(b) Fundamental theorem in space : Let be non−zero, non−coplanar vectors in space. Then any vector can be uniquily expressed as a linear combination of
There exist some unique x,y ∈ R such that
(c) If are n non zero vectors, & k1, k2, .....kn are n scalars & if the linear combination
Linearly independent vectors
(d) If are not Linearly independent then they are said to be Linearly dependent vectors & if there exists at least one kr ≠ 0 then
are said to be linearly dependent .
If then is expressed as a linear combination of vectors form a linearly dependent set of vectors. In general , every set of four vectors is a linearly dependent system. are Linearly independent set of vectors. For
Two vectors are linearly dependent ⇒ is parallel to linear dependence of Conversely if then are linearly independent .
If three vectors are linearly dependent, then they are coplanar i.e.
conversely, if then the vectors are linearly independent.
14. Coplanarity of vectors:
Four points A, B, C, D with position vectors respectively are coplanar if and only if there exist scalars x, y, z, w not all zero simultaneously such that where, x + y + z + w = 0.
15. Reciprocal system of vectors:
If are two sets of non coplanar vectors such that
then the two systems are called Reciprocal System of vectors.
16. Equation of a plane
(a) The equation represents a plane containing the point with
where is a vector normal to the plane . is the general equation of a plane.
(b) Angle between the 2 planes is the angle between 2 normals drawn to the planes and the angle between a line and a plane is the compliment of the angle between the line and the normal to the plane.
17. Application of vectors:
(a) Work done against a constant force over a displacement is defined as
(b) The tangential velocity of a body moving in a circle is given by where is the pv of the point P.
(c) The moment of about ’O’ is defined as is the pv of P wrt ’O’. The direction of is along the normal to the plane OPN such that
form a right handed system.
(d) Moment of the couple = where are pv’s of the point of the application of the forces .