May I have at least 5 examples from each types of vector

A zero vector has all its components equal to zero.

$$\vec{0} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$$ $$\vec{0} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$$

A vector with a magnitude of 1.

• For $\vec{a} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}$, its magnitude is $||\vec{a}|| = \sqrt{3^2 + 4^2} = 5$. The unit vector is $\hat{a} = \frac{1}{5} \begin{bmatrix} 3 \\ 4 \end{bmatrix} = \begin{bmatrix} \frac{3}{5} \\ \frac{4}{5} \end{bmatrix}$.
• For $\vec{b} = \begin{bmatrix} 0 \\ -7 \end{bmatrix}$, its magnitude is $||\vec{b}|| = \sqrt{0^2 + (-7)^2} = 7$. The unit vector is $\hat{b} = \frac{1}{7} \begin{bmatrix} 0 \\ -7 \end{bmatrix} = \begin{bmatrix} 0 \\ -1 \end{bmatrix}$.
• For $\vec{c} = \begin{bmatrix} 1 \\ 2 \\ 2 \end{bmatrix}$, its magnitude is $||\vec{c}|| = \sqrt{1^2 + 2^2 + 2^2} = \sqrt{1+4+4} = \sqrt{9} = 3$. The unit vector is $\hat{c} = \frac{1}{3} \begin{bmatrix} 1 \\ 2 \\ 2 \end{bmatrix} = \begin{bmatrix} \frac{1}{3} \\ \frac{2}{3} \\ \frac{2}{3} \end{bmatrix}$.

Represents the position of a point from the origin.

• Position vector of point $P(2, -1, 3)$ is $\vec{OP} = \begin{bmatrix} 2 \\ -1 \\ 3 \end{bmatrix}$.
• Position vector of point $Q(-5, 0)$ is $\vec{OQ} = \begin{bmatrix} -5 \\ 0 \end{bmatrix}$.

Represents the change in position from one point to another.

• From $A(1, 2, 3)$ to $B(4, 0, -1)$: $$\vec{AB} = \begin{bmatrix} 4-1 \\ 0-2 \\ -1-3 \end{bmatrix} = \begin{bmatrix} 3 \\ -2 \\ -4 \end{bmatrix}$$
• From $C(0, 0)$ to $D(6, -8)$: $$\vec{CD} = \begin{bmatrix} 6-0 \\ -8-0 \end{bmatrix} = \begin{bmatrix} 6 \\ -8 \end{bmatrix}$$

Two vectors are equal if they have the same magnitude and direction (i.e., identical components).

• $\vec{a} = \begin{bmatrix} 2 \\ 5 \end{bmatrix} \quad \text{and} \quad \vec{b} = \begin{bmatrix} 2 \\ 5 \end{bmatrix}$ Here, $\vec{a} = \vec{b}$.
• $\vec{p} = \begin{bmatrix} -1 \\ 0 \\ 7 \end{bmatrix} \quad \text{and} \quad \vec{q} = \begin{bmatrix} -1 \\ 0 \\ 7 \end{bmatrix}$ Here, $\vec{p} = \vec{q}$.

Has the same magnitude but opposite direction.

• If $\vec{a} = \begin{bmatrix} 3 \\ -2 \\ 1 \end{bmatrix}$, then $-\vec{a} = \begin{bmatrix} -3 \\ 2 \\ -1 \end{bmatrix}$.
• If $\vec{d} = \begin{bmatrix} -4 \\ 0 \end{bmatrix}$, then $-\vec{d} = \begin{bmatrix} 4 \\ 0 \end{bmatrix}$.

Lie on the same line or are parallel. One is a scalar multiple of the other.

• Let $\vec{a} = \begin{bmatrix} 1 \\ 2 \end{bmatrix}$. If $k=3$, then $\vec{b} = 3\vec{a} = \begin{bmatrix} 3 \\ 6 \end{bmatrix}$. $\vec{a}$ and $\vec{b}$ are collinear.
• Let $\vec{e} = \begin{bmatrix} 2 \\ -1 \end{bmatrix}$. If $k=-2$, then $\vec{f} = -2\vec{e} = \begin{bmatrix} -4 \\ 2 \end{bmatrix}$. $\vec{e}$ and $\vec{f}$ are collinear.

Lie on the same plane. Three vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ are coplanar if one can be expressed as a linear combination of the other two.

• 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, so they are coplanar.
• Let $\vec{u} = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$, $\vec{v} = \begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix}$, and $\vec{w} = \begin{bmatrix} 1 \\ 3 \\ 5 \end{bmatrix}$. Here, $\vec{w} = 1\vec{u} + 2\vec{v}$ (since $1(1)+2(0)=1$, $1(1)+2(1)=3$, $1(1)+2(2)=5$). Thus, $\vec{u}$, $\vec{v}$, and $\vec{w}$ are coplanar.