Foundations — Cross product — formula, direction (right-hand rule), torque - area calculation

Everything in the parent note is built from a small pile of ideas. This page unpacks that pile from nothing. We go in order, so each symbol you meet was already earned on the line before.

0. The rawest idea: a vector

A plain number — say the temperature $20$ — has only size. A vector has size and a heading, like "$20$ metres, north-east and slightly up".

WHY the topic needs it: the cross product eats two vectors and makes a third. If you are not fully comfortable that an arrow carries a direction, nothing after this makes sense.

See Vectors — addition, components, unit vectors for the full story of arrows.

1. Length of a vector: $A$ (no hat)

So $\vec{A}$ is the arrow; $A$ is just how long that arrow is. In the parent note the magnitude formula $|\vec{A}\times \vec{B}|=AB\sin \theta$ multiplies two lengths $A$ and $B$ — those are these hat-free numbers.

2. The three axes and unit vectors $\hat{i},\hat{j},\hat{k}$

To do arithmetic with arrows we first agree on three reference directions.

WHY the topic needs it: the parent's whole component machine is written in terms of $\hat{i},\hat{j},\hat{k}$, and the famous cycle $\hat{i}\times \hat{j}=\hat{k}$ only makes sense once you can picture these three pointers.

3. Components: $A_x,A_y,A_z$

Once we have axes, any arrow can be described by how far it reaches along each axis.

Each component is a plain (possibly negative) number: negative just means "walk the other way along that axis".

WHY the topic needs it: the determinant formula in §4 of the parent is nothing but bookkeeping on these six numbers $A_x,A_y,A_z,B_x,B_y,B_z$.

4. Tail-to-tail and the angle $\theta$

WHY the topic needs it: the magnitude $AB\sin \theta$ depends entirely on this angle. Big spread → big cross product; no spread → zero.

5. The trig switch $\sin \theta$ — and why NOT $\cos \theta$

Here is a tool entering the story, so we say exactly why this tool and not another.

Ask: "how much of $\vec{B}$ points across $\vec{A}$, not along it?" The answer is $B\sin \theta$. That across-ness is precisely the "spread" the cross product measures.

The alignment cousin $\cos \theta$ is developed in Dot product — formula, projection, work calculation.

6. Perpendicular and the right-hand rule

The cross product's direction is perpendicular to both input arrows at once. In three dimensions there are two such directions (straight out of the page, or straight into it). The right-hand rule picks which one.

WHY the topic needs it: a rotating door, a spinning wheel and a facing surface all need an axis direction. A bare number can't say "this way". The perpendicular arrow can — which is the whole reason the cross product returns a vector at all. This same handedness reappears in Angular momentum — L = r × p and Magnetic force — F = qv × B.

7. The cross symbol $\times$ and the vector-arrow output

Compare: $A\times B$ with plain numbers is just $A$ times $B$. $\vec{A}\times \vec{B}$ with arrows is a whole new arrow. The hats tell you which world you are in.

8. Area of a parallelogram and a triangle

WHY: the parent's Example 2 (triangle area) is exactly this fact in action.

9. The determinant grid

You do not need to master determinants to use this — the parent note walks the pattern — but the full mechanics live in Determinants — 3×3 expansion. The one rule to carry from here: the middle ($\hat{j}$) term always takes a minus sign.

Prerequisite map

Equipment checklist

Test yourself — cover the right side and answer before revealing.