Vector Notes for IIT JEE, Download PDF!

JEE Main Short Notes

By Neha BagriUpdated : Sep 10, 2020 , 7:45 IST
Vector Notes for IIT JEE, Download PDF!

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.

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 1 is said to be linearly independent if2

A set of vectors 1 is said to be linearly dependent if there exist scalars x1 , x2 , …, xn, not all zero such that 3

3. Collinearity

(i) Two vectors 4 and 5 are collinear ⇔  9for some scalar λ.

(ii) Three vectors4,5 and 6are collinear, if there exists scalars x, y, z such that 7 where x+y+z=0

Also the points A, B, C are collinear if8 for some scalar λ.

4. Coplanarity

(i) Three vectors s4,5 and 6 are coplanar if one of them is a linear combination of the other two if there exist scalars x and y such that 10

(ii) Four vectors 4,5 6and 11are coplanar if scalars x, y, z, w not all zero simultaneously such that

12 where x + y + z + w = 0.

5. Scalar or Dot product

The scalar product of two vectors 13is given by14 (0≤θ≤π) where θ is the angle between13

5.1 Properties of the Scalar Product


(ii) Two vectors 13 make an acute angle with each other 16, an obtuse angle if 17 and are inclined at a right angle if 18.

(iii) Projection of 19

(iv)  Projection of21

(v) Components of a vector r in the direction of a vector a and perpendicular to vector a are 22respectively.

(vi) If 23are three unit vectors along three mutually perpendicular lines, then24

(vii) 25 and 26

6. Vector or Cross Product

The vector product of two vectors 13is given by 27 where θ is the angle between the vectors and 28 is the unit vector perpendicular to 13.

6.1 Properties of the Vector Product 


(ii)30if the vectors are either along the direction or opposite in the direction.


(iv)32 then


(v) The vector perpendicular to both 13 is given by 34

(vi) The unit vector perpendicular to the plane of 13 is 35

(vii) If  three unit vectors are along three mutually perpendicular lines, then they follow the circular rule of cross product



(viii)  If 13are collinear and non-zero vectors then 30

(ix) (a) The area of a triangle if adjacent sides are 13 is given by 37

(b) The Area of a parallelogram if adjacent sides are 13 is given by 38

(c) The Area of a parallelogram if diagonals are  39is given by 40

7. Scalar Triple Product

If 41be there vectors, there called the scalar triple product of these three vectors.

Note: The scalar triple product is usually written as  42 and termed as the box a,b,c

7.1 Properties of the scalar triple product



(iii) If λ is a scalar then45

(iv) If 46

(v) The value of the scalar triple product, if two of its vectors are equal, is zero i.e., 47        

(vi) 48

(vii) The volume of the parallelepiped whose adjacent sides are represented by the vectors49

(viii) The volume of the tetrahedrane whose adjacent sides are represented by the vectors50

(ix) The volume of the triangular prism whose adjacent sides are represented by the vectors51

(x) If 52 then 41are coplanar.

(xi) If 53then 54are coplanar 

(xii) Three vectors 55form a right handed or left handed system according to as56



8. Vector Triple Product

The vector triple product of three vectors 55 is the vector59 and 60

Also, 61              

Clearly 62

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

If63 are four vectors then 64 is called the scalar product of four vectors.


This relation is known as Lagrange’s Identity.

10. Vector product of four Vectors

If 63are four vectors, the products 66are called vector products four vectors.

i.e.,67 Also, 68             

An expression for any vector, in space, as a linear combination of three non-coplanar vectors69

11. Reciprocal System of Vectors

If 55 be three non-coplanar vectors, then the three vectors 55 are defined by the equations are71 called reciprocal system of vectors to the vectors 55

11.1 Properties of Reciprocal system of Vectors





(v) The system of three mutually perpendicular unit vectors is its own reciprocal.         


(1)  The bisectors of the angles between the lines76       


‘+’ sign for internal bisector and ‘-‘ sign for external bisector.

(2) Section Formula: If 13 are the position vectors of A and B and 78 be the position vector of the point X which divides to join of A and B in the ratio m:n then79

‘+’ sign takes for internal

‘-‘ sign takes for external. 

(3) If 55 be the position vectors of ΔABC and78 be the position vector of the centroid of ΔABC. Then80

(4) The equation of a straight line

(i) Vector equation of the straight line passing through origin and parallel to 81 is given 82by where t is scalar.

(ii) Vector equation of the straight line passing through 81 and parallel to 83is given by where t is scalar.


(iii) Vector equation of the plane passing through55 is85Where s & t are scalars. 

(5) Perpendicular distance of the line 84 from the position vector of a point C represented by 86is given as


(6) Perpendicular distance of the plane i.e., 88 from the point P represented by vector a is

(7) The condition that two lines 84and 90 (where t & t1 are scalars) are coplanar and non-parallel is given by 92

(8) The shortest distance between two non-intersecting lines (skew lines ( where t & t1 are scalars ) is given by91

(9) Vector equation of the sphere with the position vector of center 81and radius p is93

(10)  Vector equation of sphere when extremities of diameter being 13 is given by94

Vector Notes for IIT JEE, Download PDF!

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Neha Bagri
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Neha Bagri