1. **Problem Statement:** We want to find the vector joining two points $P_1(x_1,y_1,z_1)$ and $P_2(x_2,y_2,z_2)$. 2. **Formula Used:** The vector from $P_1$ to $P_2$ is given by $$\vec{P_1P_2} = \vec{OP_2} - \vec{OP_1}$$ where $\vec{OP_1} = x_1\hat{i} + y_1\hat{j} + z_1\hat{k}$ and $\vec{OP_2} = x_2\hat{i} + y_2\hat{j} + z_2\hat{k}$. 3. **Substitute Position Vectors:** $$\vec{P_1P_2} = (x_2\hat{i} + y_2\hat{j} + z_2\hat{k}) - (x_1\hat{i} + y_1\hat{j} + z_1\hat{k})$$ 4. **Simplify by grouping like terms:** $$\vec{P_1P_2} = (x_2 - x_1)\hat{i} + (y_2 - y_1)\hat{j} + (z_2 - z_1)\hat{k}$$ 5. **Magnitude of the Vector:** The length of the vector is $$|\vec{P_1P_2}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$ 6. **Key Point to Remember:** Always subtract the coordinates of the starting point from the ending point to get the vector components. **Final concise formula:** $$\boxed{\vec{P_1P_2} = (x_2 - x_1)\hat{i} + (y_2 - y_1)\hat{j} + (z_2 - z_1)\hat{k}}$$ This content can be copied and pasted directly into MS Word with equations preserved.