Vectors in space

Vectors in space extend the same ideas as in the plane, but now with three coordinates. They are used to describe points, directions, and relationships in three dimensions.

 

Coordinates

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

 

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

 

 

Addition and subtraction

Two vectors in space are added by adding the corresponding coordinates:

 

$$ \large (x_1,y_1,z_1) + (x_2,y_2,z_2) = (x_1 + x_2,\; y_1 + y_2,\; z_1 + z_2) $$

 

Subtraction is done in the same way:

 

$$ \large (x_1,y_1,z_1) - (x_2,y_2,z_2) = (x_1 - x_2,\; y_1 - y_2,\; z_1 - z_2) $$

 

 

Multiplication by a number

A vector can be multiplied by a number \( \large k \) by multiplying all three coordinates:

 

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

 

 

Length

The length of a vector \( \large \mathbf{v} = (x,y,z) \) is given by:

 

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

 

 

Dot product

For two vectors \( \large \mathbf{u} = (x_1,y_1,z_1) \) and \( \large \mathbf{v} = (x_2,y_2,z_2) \), the dot product is defined as:

 

$$ \large \mathbf{u} \cdot \mathbf{v} = x_1 \cdot x_2 + y_1 \cdot y_2 + z_1 \cdot z_2 $$

 

This can also be expressed using the angle \( \large \theta \) between the vectors:

 

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

 

 

Cross product

In three dimensions, one can form the cross product of two vectors. For \( \large \mathbf{u} = (x_1,y_1,z_1) \) and \( \large \mathbf{v} = (x_2,y_2,z_2) \):

 

$$ \large \mathbf{u} \times \mathbf{v} = (y_1 \cdot z_2 - z_1 \cdot y_2,\; z_1 \cdot x_2 - x_1 \cdot z_2,\; x_1 \cdot y_2 - y_1 \cdot x_2) $$

 

The cross product is a new vector that is perpendicular to both \( \large \mathbf{u} \) and \( \large \mathbf{v} \). The length of the cross product can be interpreted as the area of the parallelogram spanned by the vectors.

 

 

Lines and planes

In space, vectors can be used to describe lines and planes.

 

A line through the point \( \large P_0(x_0,y_0,z_0) \) with direction vector \( \large \mathbf{r} = (a,b,c) \) can be written as:

 

$$ \large (x,y,z) = (x_0,y_0,z_0) + t \cdot (a,b,c), \quad t \in \mathbb{R} $$

 

A plane can be described by a normal vector \( \large \mathbf{n} = (a,b,c) \) and a point \( \large P_0(x_0,y_0,z_0) \):

 

$$ \large a(x - x_0) + b(y - y_0) + c(z - z_0) = 0 $$