Vector Geometry Formulas





Vector Geometry Formulas

 

Vector Geometry in simple words means to use vector in the geometry  inorder to make the equations and problems of geometry simpler. In order to know about vector geomerty properly you must  also know about Vector Product and Vector Formulas.

The formulas which are mostly used in vector geometry are given below:

 

Physical quantities: We have three type of physical quantities.

a. Scalar : A physical quantity having magnitude only called scalar.

e.g. Mass,Length,Time.

b.Vector: A physical Quantity having magnitude and direction is called vector e.g. Velocity,force.

c.Tensor: Not covered here

 

i. Vectors in terms of co-ordinate.

 

Let OX and OY be the X and Y axis respectively.Let P(X,Y) be a point in the plane .Join OP drawn PM \perp OX and PN \perp OY then OM=y. The directed line segment OP is known as \overrightarrow{OP} ( read \, as \, OP ) and it also devoted by (x,y) and \overrightarrow{OP} = (x,y)

 

ii.Magnitude (Modules) and direction of a vector.

 

The magnitude of a vector is it’s length;defined in 2-DD as below.

Let \overrightarrow{OP} = (x,y) and

| \overrightarrow{OP} | = \sqrt{x^2 + y^2} and direction of \overrightarrow{OP} is a unit vector parallel to \overrightarrow{OP}.

 

iii.Difference types of vectors

 

a.Unit vector : A vector having magnitude unity is called unit vector.

i.e \overrightarrow{a} is a unit vector if | \overrightarrow{a} | = 1

b.Null (zero)Vector: A vector having magnitude zero is called null(zero) vector

i.e \overrightarrow{a} is null vector of | \overrightarrow{a} | = 0

 

c.Negative vector: A vector having same magnitude as given vector \overrightarrow{OP} but opposite direction is called negative vector and is devoted by -\overrightarrow{OP} or\overrightarrow{PO} i.e \overrightarrow{OP} is negative of \overrightarrow{PO}

 

d. Like vectors: The vectors are called like vectors if they have same direction i.e. \overrightarrow{a} and \overrightarrow{b} are like vectors of \overrightarrow{a} = k:k=<0

 

e.Unlike vectors: The vectors are unlike vectors if their directions are opposite

i.e \overrightarrow{a} & \overrightarrow{b} are unlike if \overrightarrow{a} = k:k \overrightarrow{b}=<0

 

f. Equal Vectors : Two vectors are equal if they same direction and equal magnitude.\overrightarrow{a} = (a_1 , b_1 ), \overrightarrow{b} = (a_2 , b_2 ) are equal if a_1 = a_2 ; b_1 = b_2.

 

g. Localized Vectors: A vector which is passes through a given point and which is parallel to given vector is called localize vector.

 

iv.Addition of two vectors

 

Let \overrightarrow{a} = \overrightarrow{OA} = ( a_1 , a_2 ) and \overrightarrow{b} = \overrightarrow{OB} = ( b_1 , b_2 ) be two vectors.Then the sum of vectors \overrightarrow{a} & \overrightarrow{b} is denote by \overrightarrow{a} + \overrightarrow{b} and defined by \overrightarrow{a} + \overrightarrow{b} = (a_1 , a_2) + (b_1 , b_2) = ( a_1 + b_1 , a_2 + b_2)

 

v.Difference of two vectors

 

Let \overrightarrow{a} = (a_1 , a_2), \overrightarrow{b} = (b_1,b_2) then the difference of two vectors is denoted by \overrightarrow{a} - \overrightarrow{b} and defined by \overrightarrow{a} - \overrightarrow{b} = (a_1 , a_2) - (b_1 , b_2) = (a_1 - b_1 , a_2 - b_2)

 

vi. Multiplication of a vector by scalars

 

Let \overrightarrow{a} = (a_1,a_2) be a vector and k be any scalar,then the multiplication of vector by scalar k is denoted by k \overrightarrow{a} and defined by k \overrightarrow{a} = k(a_1 , a_2) = (ka_1 , ka_2)

 

vii. Collinear Vectors:

 

Any number of vectors are said to be collinear when all of them are parallel to the same live whatever the magnitude of vectors may be,otherwise non-collinear.

 

viii. Coplanar vectors

 

Let \overrightarrow{r_1} , \overrightarrow{r_2} , \overrightarrow{r_3} be set of vectors if we can write\overrightarrow{r} = a \overrightarrow{r_1} + b \overrightarrow{r_2} + c \overrightarrow{r_3} a,b,c are scalar,then the set vectors are called Coplanar.

 

ix. Unit Vector

 

Let \overrightarrow{a} = (a_1 , a_2) be a vector,then

the unit vector along \overrightarrow{a} is denoted by

\hat{a} (a,cap) and defined by \hat{a} = \dfrac{\overrightarrow{a}}{|\overrightarrow{a}|} = \dfrac{(a_1 , a_2)}{\sqrt{{a_1}^2 + {a_2}^2 }}

 

Note:

 

If \overrightarrow{a} = (a_1 , a_2 , a_3) then,

\hat{a} = \dfrac{(a_1,a_2,a_3)}{\sqrt{{a_1}^2 + {a_2}^2 + {a_3}^2}}

 

x. Direction Cosines

 

Let \overrightarrow{OP} = (x,y,z) = x \overrightarrow{i} + y \overrightarrow{j} + z \overrightarrow{k} and \alpha , \beta , \gamma be the angles made by the line OP with x,y,z axes respectively.The \cos \alpha , \cos \beta , \cos \gamma. One direction cosines of line OP and denoted by l,m,n i.e l = \cos \alpha , m = \cos \beta , n = \cos \gammaalso

l= \cos \alpha = \dfrac{x}{\sqrt{x^2 + y^2 + z^2}} , \\ m= \cos \beta = \dfrac{x}{\sqrt{x^2 + y^2 + z^2}} , \\ n= \cos \gamma = \dfrac{x}{\sqrt{x^2 + y^2 + z^2}}

Note : l^2 + m^2 + n^2 = 1 ,

 

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 denoted 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) then \overrightarrow{a} . \overrightarrow{b} = | \overrightarrow{a} | . | \overrightarrow{b} | \cos \theta where \theta being angle between the two vectors.

 

Note 1:

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

 

Note 2:

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

 

2.Properties of Scalar Product

 

i.  \overrightarrow{a} . \overrightarrow{b} =  \overrightarrow{b} . \overrightarrow{a}

 

ii. \overrightarrow{a}.( \overrightarrow{b} + \overrightarrow{c} = \overrightarrow{a} . \overrightarrow{c}

 

iii. ( \overrightarrow{a} + \overrightarrow{b} )^2 = \overrightarrow{a} ^2 + 2. \overrightarrow{a} . \overrightarrow{b} + \overrightarrow{b} ^2

 

iv. 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 , _2 , b_3) be two vectors then the cross product of \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_1b_3 - a_3b_2 , a_3b_1 - a_1b_3 , a_1b_2 - a_2b_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_2b_3 - a_3b_2) \overrightarrow{i} + (a_3b_1 - a_1b_3) \overrightarrow{j} + (a_1b_2 - a_2b_1) \overrightarrow{k}

Note 1:

\begin{vmatrix} \overrightarrow{a} \times \overrightarrow{b} \end{vmatrix} = \begin{vmatrix}\overrightarrow{a} \end{vmatrix} \begin{vmatrix} \overrightarrow{b} \end{vmatrix} \sin \theta , \theta being angle between  \overrightarrow{a} and \overrightarrow{b}

 

Note 2:

\sin \theta = \dfrac{\begin{vmatrix}\overrightarrow{a} \times \overrightarrow{b}\end{vmatrix}}{\begin{vmatrix}\overrightarrow{a}\end{vmatrix} \begin{vmatrix} \overrightarrow{b}\end{vmatrix}}

 

Note 3:

if \theta = 0, the \begin{vmatrix}\overrightarrow{a} & \times & \overrightarrow{b} \end{vmatrix} = 0 \\ i.e \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{a} \times \overrightarrow{c} \\ iv. ( n . \overrightarrow{a} ) \times \overrightarrow{b} = n ( \overrightarrow{a} \times \overrightarrow{b} = n ( \overrightarrow{a} \times \overrightarrow{b} ) = \overrightarrow{a} \times n \overrightarrow{b}

v. \overrightarrow{a} \times \overrightarrow{b} is perpendicular to both  \overrightarrow{a} and \overrightarrow{b} .

 

 \\ vi. \begin{vmatrix}\overrightarrow{a} & \times & \overrightarrow{b} \end{vmatrix} -Area of paralleogram with sides \overrightarrow{a} and \overrightarrow{b}

 vii. \dfrac{1}{2} \begin{vmatrix}\overrightarrow{a} \times \overrightarrow{b} + \overrightarrow{b} \times \overrightarrow{c} + \overrightarrow{c} \times \overrightarrow{a} \end{vmatrix}

= Area of triangle having \overrightarrow{a} , \overrightarrow{b} , \overrightarrow{c} as position vectors of verticles of a triangle.



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