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# JEE Maths Vector Algebra

## Vector Algebra PDF Notes, Important Questions and Synopsis

SYNOPSIS

Vector Algebra

1. Vector is a quantity having both magnitude and direction.
Note: A directed line segment is a vector denoted by  or simply
where  denotes ith, jth, kth components.
2. Magnitude of a vector
3.  Distance between 2 points in 3D plane is given by

d = |PQ| =

4. Internal/External division:

 Internal division Let P and Q be the two given points. Let   be the point which divides PQ internally in the ratio m : n. Then its coordinates are R = External DivisionLet P and Q be the two given points. Let    be the point which divides PQ internally in the ratio m : n.Then its coordinates areR =
5. Types of Vectors:
 Zero Vector: A vector having zero magnitude, i.e. if =  0. Also, it has no direction.   Co-initial vectors: Two or more vectors having the same initial point.   Collinear Vectors: Vectors and are said to be collinear if they are parallel to each other.  Free vectors: Vectors whose initial points are not specified. Unit Vector:A vector whose magnitude is 1, i.e. if  = 1. It is denoted by   . Equal vectors:Vectors  and   are equal if  & =  . Coplanar vectors:Vectors which are parallel or lying in the same plane are coplanar. Localised vectors:Vectors drawn parallel to a given vector, but through a specified point as the initial point. Position vector:  A vector  having O and P as its initial and terminal points, is called the position vector of point P, where O is the origin.
6. Operations on vectors:

 i. Addition of vectors:    A, B and C are three points, then    .   This is known as the triangle law of vector addition.    Also, if we have  &  , then ii. Multiplication of a vector by a scalar:     Let   be the vector and k be a scalar.       Product of   and k is  , where each  component          of  is multiplied by k.

7. Linear combination/dependence/independence

1. Linear Combination:
A vector  is said to be a linear combination of vectors  if there exist scalars such that
.
2. Linearly Independent:
A system of vectors  is said to be linearly independent if fo such that

3. Linearly Dependent:
A system of vectors
are said to be linearly dependent if there exist scalars
not all zero, such that

8. Vector Lines:
To determine vector equation of a line, we need
i. A point on the line
ii. A vector parallel to the line

9. Vector Planes:To determine vector equation of a plane, we need
1. A point on the plane
2. A vector perpendicular to the line

10. Scalar or dot product of vectors:
Scalar product of vectors and  is the projection of  over  .
Denoted by . and given by

11. Vector or cross product of vectors:
Vector product of vectors  and  is written as
and it is defined as
sin⁡θ , where n ̂ is a unit vector
along the line perpendicular to both  and .

12. Scalar triple product:
The dot product of one of the vectors with the cross product of the other two.
i.e. Scalar triple product of three vectors  and  is  .
It represents the volume of the parallelepiped.
Also, the volume of a tetrahedron is th times the volume of the parallelepiped.

13. Vector triple product:
Vector triple product of three vectors  and  is the vector

## Videos

### Concepts and application on properties of vector

Multiplication of a vector by a scalar, vector joining two points. Directio...

### Concepts on Vector cross product

Vector or cross product. Properties of vector product.

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