Working with vectors in 3-dimensional space and beyond.
The algebra of vectors in the plane can be easily generalized into the 3-dimensional space and beyond.
Everything we learned about vectors in the plane have almost verbatim generalizations to higher dimensions.
Here, of course, we allow any direction in the space.
They are represented as line segments with arrowheads, and we can write them down in coordinates in the form \[ \begin{bmatrix} a \\ b \\ c \end{bmatrix} \quad\text{or}\quad \langle a, b, c \rangle \] with (real number) components $a,b,c$ respectively.
All the basic operations generalize directly to vectors in space. For example, vector sum/difference, scalar multiplication, and vector norm have almost identical definitions/formulas for vectors in space.
\[ \begin{bmatrix} a_1 \\ b_1 \\ c_1 \end{bmatrix} + \begin{bmatrix} a_2 \\ b_2 \\ c_2 \end{bmatrix} = \begin{bmatrix} a_1 + a_2 \\ b_1 + b_2 \\ c_1 + c_2 \end{bmatrix} \]
\[ \left\| \begin{bmatrix} a \\ b \\ c \end{bmatrix} \right\| = \sqrt{ a^2 + b^2 + c^2 } \]
We still use the formulation \[ \begin{bmatrix} a_1 \\ b_1 \\ c_1 \end{bmatrix} \cdot \begin{bmatrix} a_2 \\ b_2 \\ c_2 \end{bmatrix} = a_1 a_2 + b_1 b_2 + c_1 c_2 \]
For two nonzero vectors $\mathbf{u}$ and $\mathbf{v}$, the geometric interpretation \[ \mathbf{u} \cdot \mathbf{v} = \| \mathbf{u} \| \, \| \mathbf{v} \| \, \cos(\theta) \] where $\theta$ is the angle between the two vector still holds.
The standard rectangular coordinate system for the 3-dimensional space (denoted by $\mathbb{R}^3$) uses three perpendicular axes: the $x$-axis, the $y$-axis, and the $z$-axis.
We have a very special way of arranging the axes. It is known as the right hand rule.
Position your right hand so that your fingers align with the $x$-axis, yet they would curl toward the $y$-axis. In this position, your thumb would point in the direction of the $z$-axis. This is known as the right-handed coordinates, which is incompatible with a "left-handed" coordinate system.
Sometimes it is more convenient to express vectors as combinations of "standard unit vectors".
When boldface types are difficult to write, we can also use the notations $\hat{i}$, $\hat{j}$, and $\hat{k}$ instead.
With these, we can write a vector as a "linear combination" of these standard unit vectors: \[ \begin{bmatrix} a \\ b \\ c \end{bmatrix} = a \, \mathbf{i} + b \, \mathbf{j} + c \, \mathbf{k} \]
Even though the notations $\mathbf{i},\mathbf{j},\mathbf{k}$ are pretty widely used. In certain context, you may see people use $\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3$ instead.