# Vectors Operations

Adding and Subtracting Vectors

Multiplying a Scalar by a Vector

Dot Product or Internal Product of Vectors

Cross Product or External Product of Vectors

### Adding and Subtracting Vectors

If we have two vectors,

Adding and Subtracting Vectors

Multiplying a Scalar by a Vector

Dot Product or Internal Product of Vectors

Cross Product or External Product of Vectors

If we have two vectors,

Consider the Cartesian coordinate system in space. On each of its dimensions x,y and z, we define respectively the vectors:

These vectors are independent between them and have magnitude (length) of one:

We call these vectors, unit vectors. Every other vector in this space can be constructed by adding multiples of these unit vectors. Or, a vector can be decomposed in components, in multiples of unit vectors, at all dimensions. So the vector in Image 1 can be constructed by adding:

In vector calculus del operator is a symbolic vector and is used as a partial differential operator for calculations. It is noted by the nabla symbol:

In the Cartesian coordinate system for three dimensions, del operator is defined as:

We have the right angle triangle ABC. From this orthogonal triangle the trigonometric functions are defined as follows:

The six trigonometric functions have two main characteristics, the amplitude and the frequency. The amplitude describes the height and the depth of the function, while the frequency describes how fast it is. We will use the sine function to illustrate the examples and the rest of the functions are similar. A sine function could be:

Trigonometric functions can be expressed in exponential form using Euler’s number e=2,71.

The trigonometric functions are defined from the orthogonal triangular as a ratio of its sides. In order to interpret them geometrically we will use the unit circle. The unit circle consists of a circle of radius length of one unit, which is set on the centre of the Cartesian coordinates. We will also consider a line of angle θ in respect to axis x, that begins from the origins of the Cartesian system and intersects the unit circle at the point A.

In image 1 we can see how sine, cosine and tangent are geometrically interpreted. Sine is the length of the y axis from its origin up to the projection of the point A on the y axis, cosine is the length of the x axis from its origin up to the projection of the point A on the x axis and tangent is the length of the tangent of the unit circle at the point A, from that point, up to the point that it intersects the axis x.