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Vector is one of the fundamentals for the study in other areas of mathematics and of vital importance in physics. Thus, it becomes one of the most important topics in JEE Main, JEE Advanced and other engineering entrance examinations. Every year one can expect 4-5 questions from the topic along with some questions asked in coherence with other topics. Download the Vector short notes pdf from the link given at the end of the article.
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1. Vector Quantities
Vectors are those quantities which are described by the magnitude of the quantity and its direction.
2. Linearly Independent and Dependent Vectors
A set of vectors is said to be linearly independent if
A set of vectors is said to be linearly dependent if there exist scalars x1 , x2 , …, xn, not all zero such that
3. Collinearity
(i) Two vectors and are collinear ⇔ for some scalar λ.
(ii) Three vectors, and are collinear, if there exists scalars x, y, z such that where x+y+z=0
Also the points A, B, C are collinear if for some scalar λ.
4. Coplanarity
(i) Three vectors s, and are coplanar if one of them is a linear combination of the other two if there exist scalars x and y such that
(ii) Four vectors , and are coplanar if scalars x, y, z, w not all zero simultaneously such that
where x + y + z + w = 0.
5. Scalar or Dot product
The scalar product of two vectors is given by (0≤θ≤π) where θ is the angle between
5.1 Properties of the Scalar Product
(i)
(ii) Two vectors make an acute angle with each other , an obtuse angle if and are inclined at a right angle if .
(iii) Projection of
(iv) Projection of
(v) Components of a vector r in the direction of a vector a and perpendicular to vector a are respectively.
(vi) If are three unit vectors along three mutually perpendicular lines, then
(vii) and
6. Vector or Cross Product
The vector product of two vectors is given by where θ is the angle between the vectors and is the unit vector perpendicular to .
6.1 Properties of the Vector Product
(i)
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(ii)if the vectors are either along the direction or opposite in the direction.
(iii)
(iv) then
(v) The vector perpendicular to both is given by
(vi) The unit vector perpendicular to the plane of is
(vii) If three unit vectors are along three mutually perpendicular lines, then they follow the circular rule of cross product
(viii) If are collinear and non-zero vectors then
(ix)(a) The area of a triangle if adjacent sides are is given by
(b) The Area of a parallelogram if adjacent sides are is given by
(c) The Area of a parallelogram if diagonals are is given by Everfocus download software.
7. Scalar Triple Product
If be there vectors, there called the scalar triple product of these three vectors.
Note: The scalar triple product is usually written as and termed as the box a,b,c
7.1 Properties of the scalar triple product
(i)
(ii)
(iii) If λ is a scalar then
(iv) If
(v) The value of the scalar triple product, if two of its vectors are equal, is zero i.e.,
(vi)
(vii) The volume of the parallelepiped whose adjacent sides are represented by the vectors
(viii) The volume of the tetrahedrane whose adjacent sides are represented by the vectors
(ix) The volume of the triangular prism whose adjacent sides are represented by the vectors
(x) If then are coplanar.
(xi) If then are coplanar
(xii) Three vectors form a right handed or left handed system according to as
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(xiii)
(xiv)
8. Vector Triple Product
The vector triple product of three vectors is the vector and
Also,
Clearly
Equality holds if either of the vectors is zero or all the three vectors are collinear or all three vectors are mutually perpendicular to each other.
9. The scalar product of four vectors
If are four vectors then is called the scalar product of four vectors.
This relation is known as Lagrange’s Identity.
10. Vector product of four Vectors
If are four vectors, the products are called vector products four vectors.
i.e., Also,
An expression for any vector, in space, as a linear combination of three non-coplanar vectors
11. Reciprocal System of Vectors
If be three non-coplanar vectors, then the three vectors are defined by the equations are called reciprocal system of vectors to the vectors
11.1 Properties of Reciprocal system of Vectors
(i)
(ii)
(iii)
(iv)
(v) The system of three mutually perpendicular unit vectors is its own reciprocal.
12. APPLICATION IN GEOMETRY
(1) The bisectors of the angles between the lines
Free kathleen woodiwiss ebooks. ‘+’ sign for internal bisector and ‘-‘ sign for external bisector.
(2)Section Formula: If are the position vectors of A and B and be the position vector of the point X which divides to join of A and B in the ratio m:n then
‘+’ sign takes for internal
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‘-‘ sign takes for external.
(3) If be the position vectors of ΔABC and be the position vector of the centroid of ΔABC. Then
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(4)The equation of a straight line
(i) Vector equation of the straight line passing through origin and parallel to is given by where t is scalar.
(ii) Vector equation of the straight line passing through and parallel to is given by where t is scalar.
(iii) Vector equation of the plane passing through isWhere s & t are scalars.
(5) Perpendicular distance of the line from the position vector of a point C represented by is given as
(6) Perpendicular distance of the plane i.e., from the point P represented by vector a is
(7) The condition that two lines and (where t & t1 are scalars) are coplanar and non-parallel is given by
(8) The shortest distance between two non-intersecting lines (skew lines ( where t & t1 are scalars ) is given by
(9) Vector equation of the sphere with the position vector of center and radius p is
(10) Vector equation of sphere when extremities of diameter being is given by
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