A number alone is a scalar: 40 km/h, 12 kg, three coffees. A vector carries a second fact — which way — and we draw it as an arrow. Everything in this chapter is bookkeeping for that second fact.

One arrow, three ways to name it, and each has its own symbol. Its magnitude $|\vec{v}|$ — read aloud the magnitude of v, and those upright bars mean length — is how long the arrow is, in whatever unit the story uses; a magnitude is never negative. Its components $v_x$ and $v_y$ — v-x and v-y — are the shadows it casts on the two axes: how far east, how far north, each one signed, because west is simply negative east. And its direction angle $\theta$ — the Greek letter theta, said “THAY-ta” — is measured in degrees counterclockwise from the positive x-axis, which is due east on every map in this chapter.

Two numbers describe the arrow completely, and you get to choose which pair: $(v_x,\, v_y)$ or $(|\vec{v}|,\, \theta)$. The rest of the chapter is the exchange rate between those two currencies — and the units tell you which one you are holding, because only $\theta$ wears degrees.