1. **State the problem:** We want to calculate the vector between two points in 3-dimensional space, given their coordinates. 2. **Define the points:** Suppose the first point is $P_1 = (x_1, y_1, z_1)$ and the second point is $P_2 = (x_2, y_2, z_2)$. 3. **Calculate the vector:** The vector from $P_1$ to $P_2$ is found by subtracting each coordinate of $P_1$ from the corresponding coordinate of $P_2$. 4. **Write the vector:** The vector $\vec{v}$ is $$ \vec{v} = (x_2 - x_1,\; y_2 - y_1,\; z_2 - z_1) $$ 5. **Explain:** This vector $\vec{v}$ represents the direction and distance from point $P_1$ to $P_2$ in 3D space. Each component shows how far you move along each axis. **Final answer:** To find the vector between two 3D points $P_1$ and $P_2$, compute $$ \vec{v} = (x_2 - x_1, y_2 - y_1, z_2 - z_1) $$