Multiply two vectors to get a number — the dot product. Tells you how aligned they are.
Drag the two vectors
|a| = √4²+1²
4.12
4.12
|b|
3.16
3.16
a · b = 4·1 + 1·3 = 7
angle between ≈ 57.53°
What the sign tells you
- a · b > 0 — the vectors point in broadly the same direction (angle < 90°).
- a · b = 0 — they're perpendicular.
- a · b < 0 — they point broadly opposite ways (angle > 90°). Drag the vectors above to see it flip.
Your turn
Compute (2, 3) · (4, −1) and say whether the angle is acute, right, or obtuse.
Work in physics is a dot product: W = F · d. Only the component of force along the motion counts — push perpendicular to the path and you do zero work.
Component form
Geometric
If a · b = 0, the vectors are perpendicular.