Vectors in the plane

Vectors in the plane are used to describe movements and relationships in two dimensions. They are represented as arrows in the coordinate system and can be written using their coordinates.

 

Coordinates

A vector in the plane can be described by the coordinates \( \large (x,y) \). If the vector starts at the origin and ends at the point \( \large (x,y) \), it is written as:

 

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

 

 

Addition and subtraction

Two vectors can be added by adding their corresponding coordinates:

 

$$ \large (x_1,y_1) + (x_2,y_2) = (x_1 + x_2, \; y_1 + y_2) $$

 

In the same way, vectors are subtracted by subtracting the coordinates:

 

$$ \large (x_1,y_1) - (x_2,y_2) = (x_1 - x_2, \; y_1 - y_2) $$

 

 

Multiplication by a number

A vector can be multiplied by a number \( \large k \) by multiplying each coordinate by \( \large k \):

 

$$ \large k \cdot (x,y) = (k \cdot x, \; k \cdot y) $$

 

 

Length

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

 

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

 

 

Unit vectors

A unit vector is a vector with length 1. It can be found by dividing a vector by its length:

 

$$ \large \mathbf{e} = \frac{1}{|\mathbf{v}|} \cdot \mathbf{v} $$

 

 

Dot product

For two vectors in the plane \( \large \mathbf{u} = (x_1,y_1) \) and \( \large \mathbf{v} = (x_2,y_2) \), the dot product is defined as:

 

$$ \large \mathbf{u} \cdot \mathbf{v} = x_1 \cdot x_2 + y_1 \cdot y_2 $$

 

The dot product can also be expressed using the angle \( \large \theta \) between the vectors:

 

$$ \large \mathbf{u} \cdot \mathbf{v} = |\mathbf{u}| \cdot |\mathbf{v}| \cdot \cos(\theta) $$

 

 

Projection

The projection of \( \large \mathbf{u} \) onto \( \large \mathbf{v} \) is given by the formula:

 

$$ \large \text{proj}_{\mathbf{v}}(\mathbf{u}) = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{v}|^2} \cdot \mathbf{v} $$