Length, Direction and the Dot Product

A gull swoops toward the drone. Should the warning light flash? Only if the gull is in front, where the pilot has no time to dodge. "In front of" sounds like a job for angles and trigonometry, but games answer it with two multiplications and an addition: the dot product.

The rule

Dot product sign: positive ahead, zero beside, negative behind.

Three tools, in the order you'll reach for them.

Length. An arrow's length, its magnitude, comes from Pythagoras: $\lVert \mathbf{v} \rVert = \sqrt{x^2 + y^2}$. The arrow from the drone to the crate, $(6, -8)$, is $\sqrt{36 + 64} = 10$ m long. The distance between two points is the length of the arrow between them.

Direction. Divide an arrow by its own length and you get a unit vector: length 1, pointing the same way. $(6, -8) / 10 = (0.6, -0.8)$. A unit vector separates which way from how far. To fly toward the crate at 5 m/s, the velocity is $5 \times (0.6, -0.8) = (3, -4)$.

Dot product. Multiply matching components and add: $\mathbf{a} \cdot \mathbf{b} = a_x b_x + a_y b_y$. For two unit vectors, the result is the cosine of the angle between them: 1 when they point the same way, 0 at right angles, −1 when they point opposite ways. For any two arrows, the sign alone says ahead, beside or behind.

The shortcut

"Is it in front?" needs no angles and no square roots: dot the facing direction with (target − self) and read the sign. Only for a cone, such as a camera's view, do you normalise both arrows and compare with a cosine: a target is inside a cone of half-angle $\theta$ when the dot of the two unit vectors is at least $\cos\theta$.

Two traps. Comparing a dot of un-normalised arrows with a cosine gives nonsense, because the lengths inflate the result. And for "is it within 10 m?" skip the square root entirely: compare the squared length with $10^2 = 100$.

Drills

Answer fast — trust the rule.

Remember

• Length is Pythagoras; divide by it to get a unit direction.
• Dot the facing with (target − self): positive ahead, zero beside, negative behind.
• For cones, normalise both and compare with the cosine; for distances, compare squares.