You already know how to add and scale vectors in the plane: work component by component. In space nothing changes except that a vector has three components, $\mathbf{v} = \langle v_1, v_2, v_3 \rangle$. Think of it as a displacement: $v_1$ units in the $x$-direction, $v_2$ in the $y$-direction, $v_3$ in the $z$-direction. Where you start does not matter; only the size and direction of the move do.
🔑 Key idea
From $P$ to $Q$: $\overrightarrow{PQ} = Q - P$ ("end minus start"). Vectors add and scale component by component, and
$$|\mathbf{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2}, \qquad \text{unit vector in the direction of } \mathbf{v}: \ \frac{\mathbf{v}}{|\mathbf{v}|}.$$
With $\mathbf{i} = \langle 1, 0, 0 \rangle$, $\mathbf{j} = \langle 0, 1, 0 \rangle$, $\mathbf{k} = \langle 0, 0, 1 \rangle$, every vector is $v_1\mathbf{i} + v_2\mathbf{j} + v_3\mathbf{k}$.
💡 Why it works
Adding two displacements means doing one move and then the other. The $x$-parts of the two moves simply add, and so do the $y$-parts and the $z$-parts, because the three directions are independent. The length formula is the 3-D distance formula applied to the move from the tail $(0, 0, 0)$ to the tip $(v_1, v_2, v_3)$. Multiplying by $c$ multiplies every component, so it multiplies the length by $|c|$; dividing by $|\mathbf{v}|$ therefore makes the length exactly ${}1$ without changing the direction.
Tip to tail: $\mathbf{u} + \mathbf{v}$ goes from the tail of $\mathbf{u}$ to the tip of $\mathbf{v}$. Change the components in the boxes and turn the picture to see it in 3-D.
Worked examples
✏️ Worked example: From two points
Find $\overrightarrow{PQ}$, its length, and the unit vector in its direction for $P(2, -1, 4)$ and $Q(5, 3, 4)$.
Two vectors are parallel when one is a scalar multiple of the other: $\langle 2, -4, 6 \rangle = 2\langle 1, -2, 3 \rangle$. A negative multiple points the opposite way but is still parallel.
⚠️ Common mistake
Start minus end. $P - Q$ is the vector from $Q$ to $P$, the opposite direction. Say "end minus start" out loud. A half-made unit vector. Dividing only some components by the length, or dividing by the sum of the components, gives a vector that is not length 1. Check by computing the length of your answer.
Practice
Pass with 3 right answers, the last 2 in a row. Each right answer earns 1 XP.