### Common Errors in Secondary Mathematics

Common Errors Committed  by the  Students  in Secondary Mathematics   Errors  that students often make in doing secondary mathematics  during their practice and during the examinations  and their remedial measures are well explained here stp by step.  Some Common Errors in Mathematics

### Vectors Algebra | Chapter 10 | Class XII

Vectors Algebra | Chapter 10 |  Class XII

Note: Click here to open the Assignment on Vector Algebra Chapter 10 Class 12

Topics to be discussed here

### 15) Volume of a parallelopiped using scalar Triple Product

Complete Explanation of the above mentioned topics

### 1) Scalar and Vector Quantities

All physical quantities are divided into two categories :- scalar  quantities and vector quantities.

Scalar quantities:- Those physical quantities which possess only magnitude and no direction are called scalar quantities. For example  Mass, Length, Time, Temperature , speed, Volume, Density, Work, Energy, Electric charge etc all are scalar quantities.

Vector Quantities:- Those physical quantities which possess both magnitude and direction are called vector quantities. For example : Displacement, Velocity, Momentum, Force, Weight, Acceleration etc. all are vector quantities.

Note: Quantities having magnitude and direction but do not obey the parallelogram laws of vector addition will not be treated as vectors.

For example rotation of a rigid body through a certain finite angle have both magnitude and direction but do not obey the parallelogram laws of vector addition. So it is not treated as the vector quantity.

### 2) Different types of vectors

Zero (or null) vector : A vector whose initial and final points are coincident is called zero or null vector. It has zero magnitude and no specific direction. Its direction is arbitrary.  $Zero\; vector\; is\; denoted\; by\;\; \overrightarrow{O}$

Equal Vectors : Two vectors are said to be equal if they have same magnitude and direction.

Collinear Vectors (or Parallel Vectors) : Two vectors are said to be collinear or parallel if they have same supports or parallel supports.

Co- initial Vectors : Vectors having same initial point are called co-initial vectors.

Coplanar vectors : Three or more vectors which lie in the same plane or are parallel to the same plane are called coplanar vectors. Two vectors are always coplanar.

Co-terminous vectors : Vectors having same terminal point are called co-terminous vectors.

Like vectors : Collinear vectors having same direction are called like vectors.

Negative of a vector : Two vectors of equal magnitude are said to be negative of each other  if they are of opposite direction.
Unit Vector : A vector whose magnitude is unity is called a unit vector. It is denoted by  â . Magnitude of unit vector is always 1.

### 3) Unit Vector and Method of finding unit vector

Unit Vector : A vector whose magnitude is unity is called a unit vector. It is denoted by  â . Magnitude of unit vector is always 1.
Unit vector of vector a is given by

### Now divide the given vector by its magnitude then we get a unit vector in the direction of the given vector.

If two vectors are represented in magnitude and direction by two sides of a triangle taken in the same order,  then their sum is represented by the third side taken in opposite order.

Parallelogram Law of Vector Addition : If two vectors are represented in magnitude and direction by the two adjacent sides of a parallelogram, then their sum is represented by the diagonal of the parallelogram which is coinitial with the given vectors.

or

or

### 5) Properties of vector addition

Multiplication of a vector by a scalar

### 6) Components of a Vector

Let us take the points A(1, 0, 0), B(0, 1, 0) and C(0, 0, 1) on the x-axis, y-axis and z-axis respectively. Then

The vectors    each having magnitude 1 are called unit vectors and are denoted by

If |OA| = x, |OB| = y, |OC| = z then the position vector of any vector in three dimensional plane is given by
'î' is the unit vector along x -axis,' ĵ' is the unit vector along y - axis and 'k' is the unit vector along z - axis.
If x = a,  y = b, z = c then position vector becomes
Here x component of  vector r is a, y component is b and z component is c.
a, b, c are also be called the direction ratios of the vector.

### 7) Method of finding Position Vector :

Let P be any point in the space (or plane) having co-ordinates (x, y, z) with respect to fixed point (0, 0, 0) as origin, then the vector OP is called the position vector of point P with respect to point O it is usually denoted by vector r.

Position Vector from vector A to vector B
Position vector of B - Position vector of A
Position vector  AB passing through the points A(x1, y1, z1) and B(x2, y2, z2) is given by

8) Direction angles, Direction Cosines and Direction Ratios
Direction angles:
These are the angles made by the vector with the positive direction of the axis. These are denoted by α, β, 𝜸
Direction cosines:
Cosines of the direction angles are called direction cosines.
If α, β, 𝜸 are the direction angles made by a vector with the axis then cosα, cosβ, cos𝜸 are called the direction cosines. These are also denoted by l, m, n
l = cosα,  m = cosβ,  n = cos𝜸
If l, m, n are the direction cosines of a line then   l+ m+ n2 = 1

Direction ratios:
The terms which are proportional to the direction cosines are called direction ratios. These are denoted by  (a, b, c)
For Example:
Let any vector
Its magnitude is given by
Where a, b, c are the direction ratios of the vector
The direction cosines of the given vector is given by

### 9) Section Formula and Mid Point Formula

Section Formula (For internal division)
Let A and B are two points with position vectors, vector a and vector b and let the point P with position  vector 'r'  divide the line segment AB internally  in  m : n, then

Mid Point Formula
If point P is the mid point of the line segment AB then  m = n = 1  and

Section Formula (For External  division)
Let A and B are two points with position vectors, vector a and vector b and let the point P with position  vector 'r'  divide the line segment AB Externally  in  m : n,  then

Centroid of the triangle
If A, B, C are the vertices of the triangle with position vectors, vector a, vector b, vector c, then the position vector of the centroid of the triangle is given by

### 10) Scalar or Dot product of two vectors

Scalar or dot product of two vectors, vector a and vector b is given by

where 𝞡 is the angle between two vectors, vector a and vector b.

Two vectors are perpendicular if

Two vectors are parallel if

Properties of scalar product
The scalar product is commutative i.e.

Scalar product is distributive over vector addition, i.e.

If    are two vectors then:

Angle Between two Vectors
In Vector form
If θ is the angle between  then

In Cartesian form

If   and

then

### 11) Projection of a vector on a line

then projection of    will be zero

12) Vector Product (or cross product ) of two Vectors

Vector product of two vectors  is given by

Where  is a unit vector perpendicular to both

Vector product between two vectors  is given by

Where  is a unit vector perpendicular to both

If two vectors   are perpendicular then

Because for perpendicular vectors θ = 90 and sin90o = 1

If two vectors    are parallel then

Because for parallel vectors θ = 0 and sin 0o = 0

If    then

Vector product of two vectors is not commutative or

Area of Parallelogram
If  are two vectors representing the adjacent sides of a parallelogram then
Area of parallelogram =

If  representing two diagonals of a parallelogram then
Area of parallelogram =

Area of triangle
If   are two vectors representing two sides of triangle then
Area of triangle =

Lagrange's Identity : If   are two vectors then

### 13) Scalar Triple Product of Vectors

Definition If   are three vectors then  is called the scalar triple product and is denoted by

14) Properties of Scalar Triple Product of vectors
Property 1:In scalar triple product all the three vectors are cyclically permuted

or

Property 2. Change in the order of the vectors in the scalar triple product changes the sign of the scalar triple product

Property 3. In scalar triple product the position of dot and cross can be interchanged provided that the cyclic order of the vectors remain unchanged.

Property 4. Scalar triple product of three vectors is zero if any two of them are equal.

Property 5. For any three vectors  and scalar λ, we have

Property 6 For any three vectors   and scalar l, m, n, then we have

Property 7. For any four vectors  and we have

Property 8. If    are coplanar then

### 15) Volume of a parallelopiped using scalar Triple Product

If   are co-terminous edges of the parallelopiped then
Volume of Parallelopiped =

Method of calculating the Scalar Triple Product
If

and

then

Scalar triple product of  is given by