# 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.

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