Vectors & the Dot Product
A vector is an arrow — a length and a direction. The dot product is the strange little operation that multiplies two arrows and hands back a plain number. Here's what that number is actually telling you. For Grade 12 (Ontario MCV4U and equivalents).
Two formulas, one number
The dot product has a coordinate recipe and a geometry recipe, and they always agree:
The first one is arithmetic you can do in your head. The second one says what it means: it measures how much the two arrows point the same way. Same direction → big and positive. Right angle → exactly zero. Opposite-ish → negative.
Drag the arrows 🟠🔵
Drag the tips of u and v (they snap to half-units). Watch the dot product, the angle and the green shadow — and try to make u · v exactly zero.
What to watch
The sign is the angle
You never need to compute θ to know roughly where it is: u · v > 0 means the angle is acute, < 0 means obtuse, and = 0 means perpendicular. That zero test is the single most-used fact in the course.
The green shadow is the projection
Drop a perpendicular from u's tip onto v's line: the green segment is u's shadow on v, with signed length u·v / |v|. It's how physics splits a force into "along the ramp" and "into the ramp".
Try u = (2, 1) and v = (−1, 2): the components multiply out to −2 + 2 = 0, and the arrows sit at a perfect right angle — no protractor involved. Then drag v past the perpendicular and watch the shadow flip to the negative side.