Vectors

A vector is a mathematical object that has both a direction and a magnitude. Vectors are used to describe movements, forces, and positions in both two and three dimensions.

 

You can imagine a vector as an arrow, where the length of the arrow indicates the magnitude, and the direction of the arrow indicates the direction.

 

Notation

Vectors are often written in bold \( \large \mathbf{v} \) or with an arrow above \( \large \vec{v} \).

 

If a vector goes from a point \( \large A \) to a point \( \large B \), it can be written as:

 

$$ \large \vec{AB} $$

 

Coordinate representation

In a coordinate system, a vector can be described by its coordinates. For a vector in the plane (two dimensions) with its starting point at the origin:

 

$$ \large \mathbf{v} = (x,y) $$

 

In space (three dimensions), a vector is written as:

 

$$ \large \mathbf{v} = (x,y,z) $$

 

 

Length

The length of a vector \( \large \mathbf{v} = (x,y) \) in the plane is found by:

 

$$ \large |\mathbf{v}| = \sqrt{x^2 + y^2} $$

 

For a vector in space \( \large \mathbf{v} = (x,y,z) \):

 

$$ \large |\mathbf{v}| = \sqrt{x^2 + y^2 + z^2} $$

 

 

Example

The vector \( \large \mathbf{v} = (3,4) \) has the length:

 

$$ \large |\mathbf{v}| = \sqrt{3^2 + 4^2} = 5 $$

 

This corresponds to a right triangle with legs 3 and 4 and hypotenuse 5.