# Introduction to Vectors ⬇️

Quantities that completely describe their magnitude are called scalars. Some examples of scalars include: area, length, speed or temperature. Quantities that need both a physical size, and a direction associated with it to be completely described are called vectors. Some examples of vectors include: force, velocity, displacement and acceleration.

This line has been represented as a vector. The arrow on the line indicates the direction and the length of the line its magnitude.

The vector directed from to . Which can be denoted as . We can also say that or in bold script.

**If then (the negative of vector ).**

To represent these vectors as a magnitude, which is simply the length of the line regardless of direction (can be positive or negative but always considered as positive value i.e. it is the size of something) we put the “absolute value” sign around the vector like this or .

Magnitude is a scaler for a vector and only contains information about the length and not the direction.

If two vectors appear on a graph with the same name then they need to be parallel with each other (going in the same direction) and they must be the same length, otherwise they should be considered as two different vectors. Even multiples of the vector (multiple times bigger or smaller) should be going in the same direction.

Vectors can be shown using both horizontal and vertical components contained within a 2D, cartesian (relating to Descartes and his ideas) plane.

Here . So we can use Pythagoras theorem to find its magnitude. We can say =

**If = then = **

## Extending Vectors into 3 Dimensions

Here and appears as the diagonal of a box. We can use Pythagoras again to determine its length:

=

=

**If = then = **

## Adding Vectors Together

**If and **

**then **

The corresponding vector values are simply added together to form a third vector with a combined value as shown above.

Starting from the bottom left: move along the axis 3 places, then go up the axis 2 places . From there move along the axis 2 places and down once and you’ll find that you’re in a position that has a vector of relative to your start position as shown by the green triangle in the image titled “Adding Vectors”.

**Adding two of the same vectors together:**

**NOTE:** has the same direction as but it is twice the length as we can see in the image titled “Adding ” .

**The following properties also apply**: