AM
Animated Mathematics

# Fundamental Properties of Vectors

Vectors have applications in fields that involve 2D and 3D space, examples would be plane guidance systems and computer games. Major contributions to the mathematics of vectors were made in the late 1900s by Josiah Willard Gibbs and Oliver Heaviside, from the United States and Great Britain respectively. So this is a relatively new branch of mathematics, that constructs an elegant formulation of three dimensional entities that makes 3D space accessible to scientists, engineers, computer game programmers, and anyone else who is interested (which could be you).

### Introduction

A vector is typically represented as a line segment. The direction of the vector can be thought of as a displacement going from the tail to the head. The arrow indicates the direction of travel. The vector is usually labeled by a lower case bold letter.

Two or more vectors are identical if they have the same direction and magnitude, their tails and heads need not be in the same position in space.

### Multiplying Vectors with a Scalar Quantity

A two dimensional vector can be represented by a column vector. The top number is the x displacement and the bottom number is the y displacment:

$$\mathbf{a}= \begin{bmatrix} 1 \\ 3 \end{bmatrix}$$

A vector can be multiplied by a scalar quantity, where the x and y components are multiplied by a constant. The new vector has the same direction as shown where:

$$\mathbf{b}= 2 \mathbf{a}= \begin{bmatrix} 2 \\ 6 \end{bmatrix}$$

The vector changes direction when multiplied by a negative number.

$$\mathbf{-a}= \begin{bmatrix} -1 \\ -3 \end{bmatrix}$$

### Modulus of a vector and the unit vector

For the vector $$\mathbf{a}= \begin{bmatrix} x \\ y \end{bmatrix}$$ the magnitude can be calculated using the Pythagoras theorem $$\lvert \mathbf{a} \rvert =\sqrt{x^2+y^2}$$.

For example when: $$\mathbf{a}= \begin{bmatrix} 1 \\ 3 \end{bmatrix}$$

$$\lvert \mathbf{a} \rvert =\sqrt{1^2+3^2}$$ simplifying we get $$\lvert \mathbf{a} \rvert =\sqrt{10}$$

The unit vector which has the same as direction as $$\mathbf{a}$$ is found by dividing $$\mathbf{a}$$ by it's modulus , and this vector has a magnitude of 1