Vectors

A vector is a mathematical element that contains both a direction and a magnitude. Vectors are denoted by a variable with an arrow on top, such as $\vec{r}$. A vector can for example quantify the distance between objects or different forces acting on an object.

In this article, we will consider the vectors that quantify the distance between objects with the aid of a so-called coordinate frame. Let's say a moon is orbiting around a planet in space, it can be quite useful to know where it is and in what way it is orbiting.

See the figure below for a schematic 2D representation where a planet is shown with a moon surrounding it. The position of the moon relative to the planet is described by vector $\vec{a}$.

Figure 1. Moon orbiting planet located by a 2D vector

The position of the moon can be quantified by the following expressions for $\bold{\vec{a}}$ with chosen $a_1$ and $a_2$ corresponding to the values in x -and y-direction.

In the case where a satellite is orbiting the same moon as well, specified by a distance $\vec{b}$ from the moon. The total distance between the planet and the satellite can be computed by summing the two vectors. Since the tail of $\vec{b}$ is already connected to the head of $\vec{a}$ vector addition becomes rather simple.

Figure 2. Moon orbited by satellite while orbiting the planet located by a 2D vector

In figure 2, a schematic visualization is shown of the different vectors. In the next section, the total distance between the planet and the satellite is computed and there is shown that vectors in this setup can indeed be summed.

Vector Addition

As being said the tail of vector b is already connected to vector a's head. Therefore, an immediate result of $\vec{a}+\vec{b}$ can be obtained by simply summing these two vectors. This is shown below by using concrete numbers and the grid in figure 3.

Figure 3. Quantification of the vectors

By using the grid the quantities of $\vec{a}$ and $\vec{b}$ come down to

By using addition, the total distance results in being

The found value for the total vector from the earth to the satellite can also be checked in figure 3 and there can be concluded that indeed vectors can be summed if they are connected with their heads and tails.

Now there is shown what a vector is and how vectors can be summed to identify the distances between various objects. The next readings elaborate on how to multiply vectors and how this visually should be obtained.