Vector Calculus

A scalar is a numeric quantity. A vector is a quantity that has a magnitude (the length) and the direction associated with it. A vector is represented by an arrow over a variable. For ease we will simply follow writing vector as a normal variable in here. In general we have a multi-dimensional plane ($x_1$, $x_2$, …, $x_n$). When we have two dimensional plane we write (x, y) and for three dimensional plane we refer (x, y, z). In general we discuss vectors on two or three dimensional planes.

Both scalar (like distance, speed) and vector (like displacement, velocity) fields are part of vector calculus. A scalar when zero does not have any significance while a zero vector will have significance. A zero vector may have magnitude zero but it will have some direction that will define something about that vector quantity.

For Example: if we a person has moved 3 steps N then 2 steps E and then 3 steps S, then the person has walked 8 steps and that is his distance covered. But since the person is only 2 steps to E of the starting point that means he has just moved 2 places from his basic position in the end so this makes displacement equals to two. But the direction of the vector will tell us in which way he has moved.

The magnitude of any vector is calculated by finding the square root of the sum of squares of all components of a vector:

Let us say we have u = (x, y).

Then the magnitude of u which is denoted by ||u|| or |u| =sqrt ($x^2$ + $y^2$)

Also the direction of the vector is given by $tan^{-1}$ ($\frac{y}{x}$). in case of three dimensional space the direction is associated with a unit vector in the direction of the third coordinate with n cap.

Vector Operations

Consider u and v are two vectors, then the following operations holds true.

1) Scalar multiplication:

If m is a scalar, then m . u = mu which is a vector a again.

v + u or v – u which results in a vector again.

3) Dot product:

v . u which results in a scalar number.

4) Cross product:

v x u which yields a complete vector quantity again.
All these operations have their methods and procedures to be followed for evaluation. In addition to them we also have divergence, curl and gradient. All these are full topics in themselves which require separate space to be discussed.

For Example: Let us say we have two vectors u = (3, 5, -4) and v = (4, -2, 0)

Then u + v = (3 + 4, 5 - 2, -4 + 0) = (7, 3, -4)

And u – v = (3 - 4, 5 + 2, -4 - 0) = (-1, 7, -4)

Similarly we can follow various steps associated with every operations on vectors to obtain the result.

Example

Examples on vector calculus are given below:

Example 1: Find the dot product of the given vectors given by (-6, 8) and (5, 12).

Solution:

Let vector x = (-6, 8) and vector y = (5, 12)

This implies x$_1$ = -6, x$_2$ = 5 and y$_1$ = 8, y$_2$ = 12.

The dot product of two vectors is given by the sum of the product of the respective coordinates.

So let c represent the resultant product.

Then c = a . b = -6 * 5 + 8 * 12 = -30 + 96 = 66

Example 2: Find the cross product of the following vectors: $\vec{A}$ = 2i - 3j + k and $\vec{B}$ = 4i + 5j - k.

Solution: $\vec{A}$ = 2i - 3j + k and $\vec{B}$ = 4i + 5j - k

$\vec{A}$ $\times$ $\vec{B}$ = $\begin{bmatrix} i& j & k\\ 2& -3 &1 \\ 4& 5 &-1 \end{bmatrix}$

= i(3 - 5) - j(-2 - 4) + k(10 + 12)

= -2i + 6j + 22k

Example 3: Find the dot product of $\vec{A}$ and $\vec{B}$ where, $\vec{A}$ = 3i - 6j + 4k and $\vec{B}$ = 10i − 13k

Solution:

Given $\vec{A}$ = 3i - 6j + 4k and $\vec{B}$ = 10i − 13k

Dot product of both the vectors is

$\vec{A}$ . $\vec{B}$ = (3i - 6j + 4k).(10i − 13k)

= (3 $\times$ 10) + (-6 $\times$ 0) + (4 $\times$ -13)

= 30 + 0 - 52

= -22