Vectors

VECTORS

Physical quantities are classified in to two categories i) Scalar quantities ii)Vector quantities
1. Scalar quantities or Scalars :The physical quantities which have magnitude but no direction are called scalar quantities or scalars.
Ex:Length,mass,time,area,volume,speed,energy,work,temperature etc
2.Vector quantities or Vectors:The physical quantities which posses magnitude as well as direction are called Vector quantities or Vectors.
Ex:Displacement,velocity,acceleration,momentum,force etc
Geometrical representation of vectors:A vector is geometrically represented by a directed line segment. The length of the directed line segment is called magnitude of the vector and the direction of the line segment represents the direction of the vector quantity.
Ex: If a body starts from a point O after traveling for certain time it reaches its destination point A.Then the displacement of the body is represented by the directed line segment OA . The Length of the straight line OA represents the magnitude of the displacement of the body, and the direction of the line segment from O towards A represents the direction of displacement of the body. Vector $overline{AB}$ is read as AB bar.The magnitude of $overline{AB}$ is written as $mid$ $overline{AB}$ $mid$ and read as modulus of $overline{AB}$ .

Types of Vectors :

1.Co-initial vectors: All those vectors whose initial points are same, such vectors are called co initial vectors.                                                                                                 Ex: OA,OB,OC, OD ,OE all these vectors have initial point O. Hence these are co-initial vectors.
2.Co-terminal vectors:All those vectors whose terminal (end) points are same,such vectors are     called co terminal vectors.
Ex: The terminal point of all these vectors $overline{AP}$ , $overline{BP}$ , $overline{CP}$ , $overline{DP}$    is Same point P. Hence, these vectors are called co-terminal vectors.
3. Coplanar Vectors:The vectors which lie in the same plane are called Co-planar Vectors.

Ex:a) All the vectors which lie in XY plane are called coplanar irrespective of their magnitudes
and directions. b) all the vectors which lie in YZ plane are coplanar with respect to one another.
4.Null vectors: Any vector which has direction but no magnitude is called a null vector. For a null
vector its initial point and the terminal point will be same .
Ex: $overline{AA}$ , $overline{BB}$ . $overline{OO}$ ……………….   Are null vectors.
5.Unit Vector: Any vector of unit magnitude is called unit vector.If $bar{a}$ is a vector,it’s unit vector is denoted by $hat{a}$ .This ( $hat{a}$ ) is read as a cap.
If you divide a vector with its magnitude we get the unit vector of that vector.
Therefore unit vector $hat{a}= frac{bar{a}}{ mid bar{a} mid}$
where $mid bar{a} mid$ is called the magnitude of $bar{a}$ vector.
Every vector will have its own unit vector.Unit vector of any vector will be of unit magnitude irrespective of the magnitude of the original vector.
If $bar{b}$ is a vector its unit vector will be $hat{b}$ ,the direction of $bar{b}$ and $hat{b}$ will be same.Similarly if $bar{c}$ is a vector its unit vector will be $hat{c}$ ,the direction of $bar{c}$ and $hat{c}$ will be same.
Three mutually perpendicular axes OX,OY AND OZ form the Cartesian co-ordinate system.The unit vectors along X,Y and Z axis are represented by $hat{i}$ , $hat{j}$ and $hat{k}$ respectively.
6.Like vectors : All the vectors acting in the same direction are called like vectors.
The given vectors $overline{AB}$ and $overline{DC}$
are acting in the same direction ,hence $overline{AB}$ and $overline{DC}$ are like vectors.
Note: Like vectors will be always i)parallel to one another ii) will be acting in the same direction , but iii)not necessarily be of the different magnitudes.
7. Unlike Vectors : Any two vectors parallel to one another and acting in opposite directions are called unlike vectors.

The given vectors $overline{AB}$ and $overline{CD}$ are acting in the opposite directions , hence $overline{AB}$ and $overline{CD}$ are unlike vectors.
Note: Unlike vectors will be always i)parallel to one another ii) will be acting in the opposite directions , but iii)not necessarily be of the different magnitudes.
8. Negative Vector : If $overline{AB}$ = $bar{a}$ is a vector,then $overline{DC}$ = – $bar{a}$ will be the negative vector of $bar{a}$ .
i)The original vector $bar{a}$ and it’s negative vector – $bar{a}$ will be of same magnitude i.e $mid bar{a} mid$ = $mid overline{-a} mid$ ii) they will be opposite to one another. iii) every vector will have a negative vector.
9. Axial Vectors or (One dimensional Vectors) : The vectors acting along X,Y or Z axes ( or) vectors parallel to any one of the Axis are called axial vectors (or) one dimensional vectors.
9(i). Vectors parallel to X axis : In the Fig(1) all the vectors $overline{AB}$ = $bar{a}$ , $overline{CD}$ = $bar{b}$ and
$overline{EF}$ = $bar{c}$ are parallel to X axis. These vectors $bar{a}$ , $bar{b}$ and $bar{c}$ are called axial vectors parallel to X-axis. If the magnitudes of vectors $bar{a}$ , $bar{b}$ and $bar{c}$ are $mid bar{a} mid$ =a , $mid bar{b} mid$ =b and , $mid bar{c} mid$ = c ,then we can represent the axial vectors as $bar{a}$ = a $hat{i}$ , $bar{b}$ = b $hat{i}$ and $bar{c}$ = c $hat{i}$ .
Ex: i)5 $hat{i}$ is a vector of 5units magnitude working along X-axis or parallel to X-axis in positive direction. ii) 8/5 $hat{i}$ is a vector of 8/5 units magnitude working along X-axis or parallel to X-axis in positive direction. iii) -9 $hat{i}$ is a vector of 9units magnitude working along X-axis or parallel to X-axis in negative direction. iv)a $hat{i}$ is a vector of a units magnitude working along X-axis or parallel to X-axis in positive direction.
9(ii). Vectors parallel to Y axis : In the Fig(2) all the vectors $overline{AB}$ = $bar{a}$ , $overline{CD}$ = $bar{b}$ and
$overline{EF}$ = $bar{c}$ are parallel to Y axis. These vectors $bar{a}$ , $bar{b}$ and $bar{c}$ are called axial vectors parallel to Y-axis. If the magnitudes of vectors $bar{a}$ , $bar{b}$ and $bar{c}$ are $mid bar{a} mid$ =a , $mid bar{b} mid$ =b and , $mid bar{c} mid$ = c ,then we can represent the axial vectors as $bar{a}$ = a $hat{j}$ , $bar{b}$ = b $hat{j}$ and $bar{c}$ = c $hat{j}$ .
Ex: i)3 $hat{j}$ is a vector of 3units magnitude working along Y-axis or parallel to Y-axis in positive direction. ii) 5/3 $hat{j}$ is a vector of 5/3units magnitude working along Y-axis or parallel to Y-axis in positive direction. iii) -7 $hat{j}$ is a vector of 7units magnitude working along Y-axis or parallel to Y-axis in negative direction. iv)b $hat{j}$ is a vector of b units magnitude working along Y-axis or parallel to Y-axis in positive direction.
9(iii). Vectors parallel to Z axis : In the Fig(3) all the vectors $overline{AB}$ = $bar{a}$ , $overline{CD}$ = $bar{b}$ and
$overline{EF}$ = $bar{c}$ are parallel to Z axis. These vectors $bar{a}$ , $bar{b}$ and $bar{c}$ are called axial vectors parallel to Z-axis. If the magnitudes of vectors $bar{a}$ , $bar{b}$ and $bar{c}$ are $mid bar{a} mid$ =a , $mid bar{b} mid$ =b and , $mid bar{c} mid$ = c ,then we can represent the axial vectors as $bar{a}$ = a $hat{k}$ , $bar{b}$ = b $hat{k}$ and $bar{c}$ = c $hat{k}$ .
Ex: i)6 $hat{k}$ is a vector of 6units magnitude working along Z-axis or parallel to Z-axis in positive direction. ii) 8/5 $hat{k}$ is a vector of 8/5 units magnitude working along Z-axis or parallel to Z-axis in positive direction. iii) -4 $hat{k}$ is a vector of 4units magnitude working along Z-axis or parallel to Z-axis in negative direction. iv)c $hat{k}$ is a vector of c units magnitude working along Z-axis or parallel to Z-axis in positive direction.
10.Two dimensional vectors (or) Plane vectors:The vectors acting in XY-Plane or YZ-Plane or ZX-Plane are known as two dimensional vectors or also known as vectors in a plane.
10(i).Vectors in XY-Plane: In the given Fig(i) $overline{AB}$ = $bar{a}$ is a vector working in XY-Plane.It’s X-component is $a_x$ $hat{i}$ and Y component is $a_y$ $hat{j}$ .So we can express the given vector $bar{a}$ as the sum of the component vectors.
$bar{a}$ = $a_x$ $hat{i}$ + $a_y$ $hat{j}$ .
There fore the vectors acting in XY-Plane will have only two components X-component and Y-component.These XY-plane vectors can also be represented in Cartesian co-ordinate form. $bar{a}$ =( $a_x$ , $a_y$ , 0).
Note: If X,Y co-ordinates exists(not equal to zero) and the Z co-ordinates does not exist(=0), such vectors will be in XY plane.
10(ii).Vectors in YZ-Plane: In the given Fig(ii) $overline{CD}$ = $bar{a}$ is a vector working in YZ-Plane.It’s Y-component is $a_y$ $hat{j}$ and Z component is $a_z$ $hat{k}$ .So we can express the given vector $bar{a}$ as the sum of the component vectors.
$bar{a}$ = $a_y$ $hat{j}$ + $a_z$ $hat{k}$ .

There fore the vectors acting in YZ-Plane will have only two components Y-component and Z-component.These YZ-plane vectors can also be represented in Cartesian co-ordinate form. $bar{a}$ =(0, $a_y$ , $a_z$ ).
Note: If Y,Z co-ordinates exists(not equal to zero) and the X co-ordinates does not exist(=0), such vectors will be in YZ plane.
10(iii).Vectors in XZ-Plane:
In the given Fig(i) $overline{EF}$ = $bar{a}$ is a vector working in XZ-Plane.It’s X-component is $a_x$ $hat{i}$ and Z component is $a_z$ $hat{k}$ .So we can express the given vector $bar{a}$ as the sum of the component vectors.
$bar{a}$ = $a_x$ $hat{i}$ + $a_z$ $hat{k}$ .

There fore the vectors acting in XZ-Plane will have only two components X-component and Z-component.These XZ-plane vectors can also be represented in Cartesian co-ordinate form. $bar{a}$ =( $a_x$ ,0, $a_z$ ).
Note: If X,Z co-ordinates exists(not equal to zero) and the Y co-ordinates does not exist(=0), such vectors will be in XZ plane.
11.Space Vector: If any vector possess components along all the three axes X,Y and Z such vectors are called space vectors.
$bar{a}$ = $a_x$ $hat{i}$ + $a_y$ $hat{j}$ + $a_z$ $hat{k}$ .These space vectors can also be represented in Cartesian co-ordinate form. $bar{a}$ =( $a_x$ , $a_y$ , $a_z$ ).
In the given fig $overline{OE}$ is a space vector .Where OC = $a_x$ , OA = $a_y$ and OG = $a_z$ .Hence $overline{OE}$ = $bar{r}$ = $a_x$ $hat{i}$ + $a_y$ $hat{j}$ + $a_z$ $hat{k}$ .

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