Vectors & Spaces · Lesson 1 of 3 · ~6 min

Vectors in the plane

Predict

$\mathbf{u} = (2, 1)$ and $\mathbf{v} = (1, 2)$. You walk leg $\mathbf{u}$, then leg $\mathbf{v}$, and draw one arrow from the origin to where you stop. Where does $\mathbf{u} + \mathbf{v}$ land?

Hold that thought. The instrument opens with exactly these two arrows — check the u + v readout.

Try

Try itDrag the tips of u and v. The dashed parallelogram redraws and the arrow labeled u+v rides its diagonal; the u + v readout tracks the sum. Then sweep the c · u slider from $2$ to $-2$ and watch the marker c·u.

Notice

With the opening pair the readout shows $\mathbf{u} + \mathbf{v} = (3.00, 3.00)$: first components $2 + 1$, second components $1 + 2$. Drag $\mathbf{u}$ to $(3, -1)$ and the sum snaps to $(4.00, 1.00)$ — the rule never changes. And the marker c·u never leaves the dotted line through $\mathbf{u}$: with $\mathbf{u} = (2, 1)$ and $c = -1.00$ it sits at $(-2, -1)$, the exact reverse.

Explain

A vector is an ordered pair $\mathbf{v} = (v_1, v_2)$ — its components — drawn as an arrow from the origin. There are only two legal moves, and both act one component at a time:

\mathbf{u} + \mathbf{v} = \begin{pmatrix} u_1 + v_1 \\ u_2 + v_2 \end{pmatrix}, \qquad c\,\mathbf{u} = \begin{pmatrix} c\,u_1 \\ c\,u_2 \end{pmatrix}

Each has a picture. Addition is tip-to-tail: slide a copy of $\mathbf{v}$ to the tip of $\mathbf{u}$, and the sum runs to the far corner of the parallelogram — the same corner either walking order reaches. Scaling keeps the direction and rescales the length: $c > 1$ stretches, $0 < c < 1$ compresses, $c < 0$ flips through the origin, and $c\,\mathbf{u}$ never leaves the line through $\mathbf{u}$. The arrow also has a size: by Pythagoras its magnitude is $\|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2}$.

Apply

Checkpoint — prove it to yourself

Walk the legs in the other order: $\mathbf{v} = (1, 2)$ first, then $\mathbf{u} = (2, 1)$. Where does $\mathbf{v} + \mathbf{u}$ land?

With $\mathbf{u} = (2, 1)$, where does the marker sit when the slider reads $c = -1.5$?

Transfer: combine both moves. With $\mathbf{u} = (2, 1)$ and $\mathbf{v} = (1, 2)$, compute $2\mathbf{u} - \mathbf{v}$.

Key idea

Every fact about vectors reads twice: once as algebra on components, once as geometry on arrows. Componentwise addition and the parallelogram are the same statement in two forms — check each against the other.

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