The dot product asks how much two arrows AGREE. The cross product asks the opposite question — how much they SPREAD — and its magnitude is $|\vec{a}\times\vec{b}| = |\vec{a}|\,|\vec{b}|\sin\theta$, read aloud the magnitude of a cross b equals the magnitude of a times the magnitude of b times sine theta, where $\theta$ is again the angle between them, from $0^\circ$ to $180^\circ$. Cosine for the dot, sine for the cross: get that pair the wrong way round and every answer in this lesson inverts.

And the number has a picture. Lay the two arrows tail to tail and complete the parallelogram: $|\vec{a}\times\vec{b}|$ IS its area. Parallel arrows sweep nothing, $\sin 0^\circ = 0$, zero area — the mirror image of the dot product vanishing at $90^\circ$. Given components instead of an angle, the same area comes from the determinant: $A = \left|a_x b_y - a_y b_x\right|$ — cross-paired and SUBTRACTED, the opposite of the dot's like-paired sum, and the outer bars are absolute value, because an area is never negative. Two arrows from one corner also cut off a triangle, and it is exactly half: $A = \tfrac{1}{2}\left|a_x b_y - a_y b_x\right|$. Read the question twice — parallelogram or triangle decides whether that $\tfrac{1}{2}$ belongs in your answer.