A vector is a quantity with magnitude and direction, and magnitude of a vector is known as length. Vector is represented by a directed line segment. The direction of the vector is indicated by an arrow pointing from the tail to head. If tail is at point A and head is at point B then the vector from A to B is written $\vec{AB}$  

Examples: Acceleration, momentum, force magnetic field etc.

A vector is a combination of three things.
  1. Magnitude which is a positive number,
  2. Direction in space and
  3. The idea of direction.
A vector is represented by a straight line which will have an starting point and an ending point.
A two-dimensional vector will be an ordered pair of real numbers v = (a, b) where a and b are the components of v.

The length of the vector v = (a, b) is defined as |v| = v = $\sqrt{(a^{2} + b^{2})}$
Let r(t) be a differentiable vector valued function and let T(t) be the unit tangent vector. Unit normal vector N(t) is defined as

N(t) = $\frac{T'(T)}{||T'(t)||}$
Vector analysis is concerned with differentiation and integration of vector fields in $R^{3}$ . Vector analysis is also known as vector calculus and was invented by Willard Gibbs and Oliver Heaviside. In vector calculus the basic objects are scalar and vector fields which are later combined into various operations and integrated. Vector analysis also generalizes results for curves, surfaces and volumes in $R^{n}$.

For example: Two forces act on an object the sum of the resulting force will depend on both the direction and magnitude.
Vector equation is formed by a collection of elements called vectors which may be added together or multiplied. Vector equation consists of two vectors - a position vector which will be from origin to any point on the line and a direction vector which describes gradient (Simplest vector operator is the gradient and is normally written as an inverted delta with a vector line over it. It can also be written as grad). Vector equation is usually found for a line.
For a vector quantity it is necessary to specify both magnitude and direction, which will be quantified by a number. Vector quantities of same type (any number) can be combined to give four basic vector operations.
  • Vector addition.
  • Vector subtraction.
  • Vector dot product.
  • Vector cross product.
Vector dot product is also known as scalar product it is an operation that takes two vectors, 'multiplies' them together and the result is a scalar.

Dot product of two vectors is defined as
v = a$\vec{i}$ + b$\vec{j }$ and w = c$\vec{i }$ + d$\vec{j}$
v.w = ac + bd ($\because$ i . i = j . j = 1)
Dot product is commutative, distributive and associative.
Vector cross product is a binary operation on two vectors in three-dimensional space. Also known as vector product is an operation on two vectors. While finding the cross product of two vectors the third vector will be perpendicular to the plane in which the first two lie. Applications of vector can be seen in mathematics and engineering.

Let l = a$\vec{i }$ + b$\vec{j }$ + c$\vec{k}$ and m = d$\vec{i}$ + e$\vec{j }$+ f$\vec{k}$ be vectors then the cross product l $ \times$ m is found by the determinant of the matrix.
$\begin{Bmatrix}
i& j & k\\
a& b& c\\
d& e& f
\end{Bmatrix}$
Vector space is a collection of elements called vectors which may be added together and multiplied.
A vector space V is a set which should satisfy the following for all u, v and w in V and scalars c and d.

Closure u + v exists in V
Commutative u + v = v + u
Associative (u + v) + w = u + ( v + w)
Additive Identity u + 0 = u
Inverse u + (-u) = 0
Closure under scalar multiplication cu exists in V
Distributive left c( u + v) = cu + cv
Distributive right
(c + d) u = cu + du
Multiplicative Identity 1u = u
Multiplicative Associative property c(du) = (cd)u

Vector field is a collection of arrows with a given magnitude and direction each attached to a point in the plane. The applications of vector field can be seen in differential and integral calculus and representation of vector field depends on the coordinate system.
Definition: Vector field is a vector having scalar field and the function of variables. The following notation is used to represent vector field.
v(x, y, z) = $v_{1}$(x, y, z), $v_{2}$(x, y, z), $v_{3}$(x, y, z)

The vector projection is the unit vector of $\vec{v}$ by the scalar projection of u on v.
The formula for vector projection is

proj$_{v}$ u = $\frac{\vec{u}.\vec{v}}{\vec{|v|}}$$\frac{\vec{v}}{\vec{|v|}}$
=$\frac{\vec{u}.\vec{v}}{\vec{|v|}^{2}}$$\vec{v}$
The scalar projection of u on v is the magnitude of the vector projection of u on v.
Curl
$\vec{\bigtriangledown}\times (\vec{L}+\vec{M})=\vec{\bigtriangledown}\times\vec{L}+\vec{\bigtriangledown}\times\vec{M}$
$\vec{\bigtriangledown}\times(c\vec{L})=c\vec{\bigtriangledown}\times\vec{L}$ for any constant c
$\vec{\bigtriangledown}\times(l\vec{L})= l\vec{\bigtriangledown}\times\vec{L} +\vec{\bigtriangledown}l\times \vec{L}$
$\vec{\bigtriangledown}\times(\vec{L}\times \vec{M}) = \vec{L}(\vec{\bigtriangledown}.\vec{M})-(\vec{\bigtriangledown}. \vec{L})\vec{M}+(\vec{M}.\vec{\bigtriangledown})\vec{L}-(\vec{L}.\vec{\bigtriangledown})\vec{M}$

Divergence
$\vec{\bigtriangledown} . (\vec{L}+\vec{M})=\vec{\bigtriangledown}.\vec{L}+\vec{\bigtriangledown}.\vec{M}$
$\vec{\bigtriangledown} . (c\vec{L})=c\vec{\bigtriangledown}.\vec{L}$ for any constant c
$\vec{\bigtriangledown} . (l\vec{L})= l\vec{\bigtriangledown}.\vec{L} + \vec{L}.\vec{\bigtriangledown}$l
$\vec{\bigtriangledown} . (\vec{L}\times \vec{M}) = \vec{M}. (\vec{\bigtriangledown} \times \vec{L})-\vec{L}.(\vec{\bigtriangledown} \times \vec{M})$

Degree two

$\vec{\bigtriangledown }.(\vec{\bigtriangledown }\times\vec{L})=0$
$\vec{\bigtriangledown }\times(\bigtriangledown \vec{l})=0$
$\vec{\bigtriangledown }.(\vec{\bigtriangledown }l\times \vec{\bigtriangledown }m)=0$

Gradient
$\vec{\bigtriangledown} (l+m)=\vec{\bigtriangledown}l+\vec{\bigtriangledown}m$
$\vec{\bigtriangledown}(cl)=c\vec{\bigtriangledown}$l for any constant c
$\vec{\bigtriangledown}(lm)= l\vec{\bigtriangledown}m + m\vec{\bigtriangledown}l$
$\vec{\bigtriangledown}(\vec{L}.\vec{M}) = \vec{L} \times (\vec{\bigtriangledown} \times \vec{M})-(\vec{\bigtriangledown} \times \vec{L})\times \vec{M}+(\vec{M}.\vec{\bigtriangledown} )\vec{L}+(\vec{L}.\vec{\bigtriangledown} )\vec{M}$

Laplacian

$\vec{\bigtriangledown ^{2}}(l+m)=\vec{\bigtriangledown ^{2}}l+\vec{\bigtriangledown ^{2}}$m
$\vec{\bigtriangledown ^{2}}(cl)=c\vec{\bigtriangledown ^{2}}$l for any constant c
$\vec{\bigtriangledown ^{2}}(lm)=l\vec{\bigtriangledown ^{2}}m+2\vec{\bigtriangledown}l.\vec{\bigtriangledown}m+m\vec{\bigtriangledown ^{2}}$l
Zero vector or null vector is a vector which has zero magnitude and an arbitrary direction. It is represented by $\vec{0}$. If a vector is multiplied by zero, the result is a zero vector. All the components will be zero and it is the additive identity of the additive group of vectors.
Magnitude of the vector $\vec{AB}$ is the length of the line segment $\bar{AB}$. Denoted by $\vec{|AB|}$.
Magnitude will be a positive number or zero and magnitude of vectors cannot be added algebraically.If $\vec{u}$ = ($u_{1}$,$u_{2}$)
$\bar{|u|}$ = $\sqrt{u_{1}^{2}+u_{2}^{2}}$

Solved Example

Question: Find the length of the vector 3i + 2j + 1 k.
Solution:
 
Let x = 3, y = 2,  z = 1
Formula to find the length of the vector is $\sqrt{x^{2}+ y^{2}+ z^{2}}$
Plug in the values
$\sqrt{3^{2}+ 2^{2}+ 1^{2}}$ = 3.74
 

Gradient of f is the unique vector field whose dot product with any vector v at each point x is the directional derivative of f along v. Gradient of a scalar function f($x_{1},x_{2}, ....., x_{n}$) is denoted by $\bigtriangledown$f or $\vec{\bigtriangledown}$f. Gradient represents the steepness and direction of slope. $\bigtriangledown$f (nabla) is the vector differential operator.

($\bigtriangledown$f(x)).v = $D_{v}$f(x)
Column vector is an m $\times$ 1 matrix. That is it will have single column of m elements.
$\begin{Bmatrix}
x_{1}\\
x_{2}\\
.\\
.\\

x_{m}\\
\end{Bmatrix}$
A vector perpendicular to a given vector 'a' is a vector a $^{\perp}$. 'a' and a$^{\perp}$ will form a right angle(90$^{0}$).

In a plane if there are two vectors one will be rotated (90$^{0}$) counterclockwise and other will be rotated (90$^{0}$) clockwise.
Vector triple product is the cross product of one vector with the cross product of other two.

Product of three vectors will give triple product.

$a \times(b \times c)$ =(a . c) b - (a . b) c and

$(A \times B)\times C$ = $-C \times (A\times B)$
= - A ( B . C ) + B ( A . C )
If a vector is tangent to a given curve then it is said to be a tangent vector to that curve.
Let r(t) be a differentiable vector valued function and v(t) = r'(t) be the velocity vector. Then the unit tangent vector will be in the direction of velocity vector and is defined as

T(t) = $\frac{v(t)}{||v(t)||}$

Solved Examples

Question 1: Find the length of the vector 7i + 2j + 6k.
Solution:
 
Let x = 7, y = 2, z = 6
Formula to find the length of the vector is $\sqrt{x^{2}+ y^{2}+ z^{2}}$
Plug in the values
$\sqrt{7^{2}+ 2^{2}+ 6^{2}}$ = 9.43
 

Question 2: Calculate the vector projection of $\vec{u}$ = 7$\vec{i}$ - 3$\vec{j}$ on the vector $\vec{v}$ = 6$\vec{i}$ + 6$\vec{j}$.

Solution:
 
$\vec{u}$ = 7$\vec{i}$ - 3$\vec{j}$ = 7(1, 0) - 3(0, 1)
= (7, 0) + (0, -3)  = (7, -3)

$\vec{v}$ = 6$\vec{i}$ +6 $\vec{j}$ = 6(1, 0) + 6 (0, 1)
= (6, 0) + (0, 6)  = (6, 6)

The formula to find vector projection is given by
proj$_{v}$ u = $\frac{\vec{u}.\vec{v}}{\vec{|v|}^{2}}$$\vec{v}$

$\frac{7.6 - 3.6}{36+36}$ (6, 6)

= $\frac{42 - 18}{72}$ (6, 6)

= ($\frac{144}{72}$, $\frac{144}{72}$)

= (2, 2)