Vector Formulas





Vector Formulas

A vector can also be defined as an element of a vector space.  Vectors are sometimes referred to by the number of coordinates they have, so a 2-dimensional vector  is often called a two-vector, an n-dimensional vector is often called an n-vector, and so on.

The important formulas of vectors are given below:

 

1. The position vector of any point p(x,y) is

op = \dbinom{x}{y} or OP = ( x,y ).

 

2.The magnitude of position vector
OP = \sqrt{x^2 + y^2} and direction \tan \theta = \dfrac{y}{x}

 

3. The unit vector = \dbinom{1}{0} where the magnitude of unit vector is 1

Or,the unit vector = \dfrac{vector}{its modulus} = \dfrac{ \overrightarrow{a}}{ |\overrightarrow{a}| }

 

4.The two vectors \overrightarrow{a} and \overrightarrow{b} are parallel if \overrightarrow{a} = k \overrightarrow{b} and \overrightarrow{b} = m \overrightarrow{a} where k and m are the scalars.

 

5.If \overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC} then \overrightarrow{AC} is the result vector which is the triangle law of vector addition.

 

6. The scalar or dot product of any two vectors \overrightarrow{a} . \overrightarrow{b} = | \overrightarrow{a} | | \overrightarrow{b} | cos\theta.

 

7. The angle between two vectors is  \cos \theta = \dfrac{\overrightarrow{a} . \overrightarrow{b}} {| \overrightarrow{a} | | \overrightarrow{b} |}

 

8. \overrightarrow{a} = x_1 i + y_1 j and \overrightarrow{b} = x_2 i + y_2 j , then :
\overrightarrow{a} . \overrightarrow{b} = x_1 . x_2 + y_1 . y_2 where i.j = j.1 = 0

 

9. If the position vector of A is \overrightarrow{a} , position vector of point B is \overrightarrow{b} and position vector of mid-point M is m then \overrightarrow{m} = \dfrac{\overrightarrow{a} + \overrightarrow{b}}{2}

 

10. If the point P divides Ab internally in the ratio m:n then position vector of P is given by \overrightarrow{p} = \dfrac{n \overrightarrow{a} + m \overrightarrow{b}}{m + n} which is a section formula.

 

11.If P divides AB externally in the ratio m:n then \overrightarrow{p} = \dfrac{m \overrightarrow{b} - n \overrightarrow{a}}{m - n}

PRODUCT OF TWO VECTORS

 

1.Scalar Product ( dot product )

 

Let \overrightarrow{a} = (a_1,a_2) , \overrightarrow{b} = (b_1,b_2) then dot product of \overrightarrow{a} & \overrightarrow{b} is devoted by \overrightarrow{a}.\overrightarrow{b} read as \overrightarrow{a} dot \overrightarrow{b} and defined by \overrightarrow{a}.\overrightarrow{b} = a_1 b_1 , a_2 b_2

 

Note:

if\overrightarrow{a} = (a_1,a_2,a_3),\overrightarrow{b} = (b_1,b_2,b_3) \\ \overrightarrow{a} . \overrightarrow{b} = a_1b_1 + a_2b_2 +a_3b_3

 

OR

 

The scalar product of \overrightarrow{a} & \overrightarrow{b} is devoted by \overrightarrow{a} . \overrightarrow{b} ,
\overrightarrow{a} . \overrightarrow{b} |\overrightarrow{a}|.|\overrightarrow{b}| \cos \theta where \theta being  angle between \overrightarrow{a} & \overrightarrow{b}

 

Note:1

\cos \theta = \dfrac{\overrightarrow{a} . \overrightarrow{b}}{| \overrightarrow{a} | | \overrightarrow{b} |}
Note:2

\overrightarrow{a} & \overrightarrow{b} are perpendicular if \theta = 90^o
i.e \overrightarrow{a} . \overrightarrow{b} |\overrightarrow{a}|.|\overrightarrow{b}| \cos 90^o or \overrightarrow{a} . \overrightarrow{b} = 0

 

2.Properties of Scalar Product
i. \overrightarrow{a} . \overrightarrow{b} = \overrightarrow{b} . \overrightarrow{a}.
ii. m \overrightarrow{a} . n \overrightarrow{b} - mn \overrightarrow{a}. \overrightarrow{b} = \overrightarrow{a} mn \overrightarrow{b}.
iii. \overrightarrow{a}( \overrightarrow{b} + \overrightarrow{c} ) = \overrightarrow{a} . \overrightarrow{c}
iv. ( \overrightarrow{b} + \overrightarrow{c} )^2 = \overrightarrow{a}^2 + 2.\overrightarrow{a}.\overrightarrow{b} + \overrightarrow{b}^2
v. If \overrightarrow{i} = (1,0,0): \overrightarrow{j} = (0,1,0), \overrightarrow{k} = (0,0,1)then \overrightarrow{i} . \overrightarrow{j} = \overrightarrow{j} . \overrightarrow{k} = \overrightarrow{k} . \overrightarrow{i} = \overrightarrow{i} . \overrightarrow{k} = \overrightarrow{j} . \overrightarrow{k} = 0

 

3.Vector (cross) Product of two vectors.
Let \overrightarrow{a} = (a_1 , a_2 , a_3 ), \overrightarrow{b} = (b_1 , b_2 , b_3 ) be two vectors then the cross product of \overrightarrow{a} \times \overrightarrow{b}is devoted by\overrightarrow{a} \times \overrightarrow{b} and defined by

\overrightarrow{a} \times \overrightarrow{b} = (a_1 , a_2 , a_3 ) \times (b_1 , b_2 , b_3 )

 

 = \begin{pmatrix} a_1 & a_2 & a_3 & a_1 & a_2 \\ b_1 & b_2 & b_3 & b_1 & b_2 \end{pmatrix} = ( a_2 b_3 - a_3 b_2 , a_3 b_1 - a_1 b_2 - a_2 b_1 )

 

We can define in terms of determinants as follows

 

\overrightarrow{a} \times \overrightarrow{b} = \begin{vmatrix} \overrightarrow{i} & \overrightarrow{j} & \overrightarrow{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}

 

= ( a_2 b_3 - a_3 b_2 ) \overrightarrow{i} + ( a_3 b_1 - a_1 b_3 ) \overrightarrow{j} + ( a_1 b_2 - a_2 b_1 ) \overrightarrow{k}

Note:1               | \overrightarrow{a} \times \overrightarrow{b} | = | \overrightarrow{a} | \times | \overrightarrow{b} | \sin \theta , \theta being angle between
\overrightarrow{a} & \overrightarrow{b}
Note:2               \dfrac{| \overrightarrow{a} \times \overrightarrow{b}| }{| \overrightarrow{a} | | \overrightarrow{b} | \sin \theta }
Note:3               If \theta = 0 , the | \overrightarrow{a} \times \overrightarrow{b} | = 0 i.e \overrightarrow{a} \times \overrightarrow{b} = 0 and \overrightarrow{a} & \overrightarrow{b} are parallel if \overrightarrow{a} \times \overrightarrow{b} = 0.

 

4. Properties of cross product

 

i. \overrightarrow{a} \times \overrightarrow{b} = 0
ii. \overrightarrow{a} \times \overrightarrow{b} = \overrightarrow{b} \times \overrightarrow{a}
iii. \overrightarrow{a} \times ( \overrightarrow{b} + \overrightarrow{c} ) = \overrightarrow{a} \times \overrightarrow{b} + \overrightarrow{a} \times \overrightarrow{c}
iv. ( n \overrightarrow{a} ) \times \overrightarrow{b} = n( \overrightarrow{a} \times \overrightarrow{b} ) \times n \overrightarrow{b}
v. \overrightarrow{a} \times \overrightarrow{b} is perpendicular to both \overrightarrow{a} and \overrightarrow{b}
vi. | \overrightarrow{a} \times \overrightarrow{b} | is a Area of paralelogram with sides  \overrightarrow{a} and  \overrightarrow{b}
vii. \dfrac{1}{2} | \overrightarrow{a} \times \overrightarrow{b} + \overrightarrow{b} \times \overrightarrow{c} + \overrightarrow{c} \times \overrightarrow{a} | = area of triangle having \overrightarrow{a} , \overrightarrow{b} , \overrightarrow{c} as position vectors of vertices of a triangle.



Related posts:

  1. Maths Formulas for Physics Maths Formulas for Physics. List of mathematical formulas used in...
  2. Vector product ( Multiplying vectors ) Multiplying vectors. Multiplying vectors with scalars , and vectors. Vector...
  3. Unit Vector Unit Vector. What is unit vector? What are standard unit...
  4. Components of vectors Components of vectors , components of vectors in two and...
  5. Adding vectors by components Adding vectors by components. Vector sum by component method....