May I have mathematical examples of each type of vector?

A zero vector in two dimensions is represented as:

In three dimensions:

If we have a vector $\vec{a} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}$, its magnitude is $||\vec{a}|| = \sqrt{3^2 + 4^2} = \sqrt{9+16} = \sqrt{25} = 5$.
The unit vector in the direction of $\vec{a}$ is:

The position vector of a point $P(x, y, z)$ from the origin $O(0, 0, 0)$ is:

For example, the position vector of point $P(2, -1, 3)$ is:

If point $A$ is $(x_1, y_1, z_1)$ and point $B$ is $(x_2, y_2, z_2)$, the displacement vector from $A$ to $B$ is $\vec{AB} = \vec{OB} - \vec{OA}$.
For example, if $A(1, 2, 3)$ and $B(4, 0, -1)$:

Two vectors $\vec{a}$ and $\vec{b}$ are equal if their corresponding components are equal.

Here, $\vec{a} = \vec{b}$.

If $\vec{a} = \begin{bmatrix} 3 \\ -2 \\ 1 \end{bmatrix}$, its negative vector is:

Two vectors $\vec{a}$ and $\vec{b}$ are collinear if $\vec{b} = k\vec{a}$ for some scalar $k$.
Let $\vec{a} = \begin{bmatrix} 1 \\ 2 \end{bmatrix}$. If $k=3$, then:

Vectors $\vec{a}$ and $\vec{b}$ are collinear.

Three vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ are coplanar if one can be expressed as a linear combination of the other two (i.e., $\vec{c} = x\vec{a} + y\vec{b}$ for scalars $x, y$).
Let $\vec{a} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$, $\vec{b} = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}$, and $\vec{c} = \begin{bmatrix} 2 \\ 3 \\ 0 \end{bmatrix}$.
Here, $\vec{c} = 2\vec{a} + 3\vec{b}$. All three vectors lie in the $xy$-plane (where $z=0$), so they are coplanar.