Dot Product and Cross Product

Two ways of multiplying vectors and what each one rewards: the dot product peaks when vectors are parallel and vanishes at right angles, while the cross product does the opposite - with everyday examples and the physics that follows.

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Dot Product and Cross Product — Moosa Academy

Vectors can be multiplied in two entirely different ways, and the two are opposites. One rewards vectors that line up; the other rewards vectors that stand at right angles.

Concept The dot product rewards alignment

Written \vec{A} \cdot \vec{B} , the dot product is largest when the two vectors point the same way.

Parallel vectors (0°) — the greatest result.
As the angle grows — the result shrinks.
Perpendicular vectors (90°) — the result is zero.
Example Pushing a car
force along the path angle 0° force straight down angle 90° maximum zero
Push a car along the road it is travelling and every bit of your effort counts. Push straight down instead and the car moves nowhere — the effort in the direction of travel is zero.
Concept The cross product rewards perpendicularity

Written \vec{A} \times \vec{B} , the cross product behaves in exactly the opposite way.

Perpendicular vectors (90°) — the greatest result.
As the angle closes — the result shrinks.
Parallel vectors — the result is zero.
Example Pushing a door
push at 90° push along the door maximum zero
A door opens best when you push square on to it. Slide your hand up or down the surface instead and nothing happens at all.

The same pattern appears in physics: the magnetic force on a current is greatest when the current runs perpendicular to the magnetic field. That is a cross product at work.

Note Telling them apart

Ask one question: when is the effect greatest?

Greatest when the vectors are parallel — it is a dot product, written with \cdot .
Greatest when the vectors are perpendicular — it is a cross product, written with \times .

The symbols are easy to mix up in handwriting, so read them carefully — a dot and a cross mean completely different operations.

Summary
  1. The dot product A · B is greatest for parallel vectors and zero at 90°.
  2. The cross product A × B is greatest at 90° and zero for parallel vectors.
  3. Pushing a car is a dot product; opening a door is a cross product.
  4. Magnetic force is largest when current and field are perpendicular.
  5. Ask when the effect peaks — parallel means dot, perpendicular means cross.
231. Arithmetic Sequences and Series
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