Use the cross-product direction and laws

Use the cross-product direction and laws

Keep the factor order visible. For nonparallel vectors, curl from the first towards the second with your right hand to choose the normal direction. The cross product is a vector; its magnitude gives a parallelogram area. To find a perpendicular length, cross with a unit direction.

For two nonparallel nonzero vectors in three dimensions, the vector product, or cross product, is

$$\mathbf a\times\mathbf b=(|\mathbf a||\mathbf b|\sin\theta)\widehat{\mathbf n},\qquad0<\theta<\pi.$$

Here $\widehat{\mathbf n}$ is perpendicular to both vectors. Curl the fingers of your right hand from $\mathbf a$ towards $\mathbf b$ through the smaller angle; your thumb selects $\widehat{\mathbf n}$.

The result is a vector. Reversing the order reverses the normal.

The source’s hand illustration supplies this curl/thumb rule; its accompanying physics motivation mentions induced current in a moving wire, generators, dams and windmills and uses the name Fleming’s right-hand rule.

For the mathematics here, apply the stated curl from the first vector to the second convention.

If either vector is zero, the cross product is $\mathbf0$. Parallel nonzero inputs also give zero. In these cases there is no nonzero cross-product direction to normalise; the unit normal in the displayed definition is used only in the nonparallel case.

Apply the cross-product laws

The six source properties and their proof roles are:

1. Magnitude. Taking the norm of $(|\mathbf a||\mathbf b|\sin\theta)\widehat{\mathbf n}$ gives $|\mathbf a\times\mathbf b|=|\mathbf a||\mathbf b|\sin\theta$, since $|\widehat{\mathbf n}|=1$ and $\sin\theta\ge0$ for $\theta\in[0,\pi]$. No absolute value around the sine is needed on this interval.
2. Parallel iff zero product, for nonzero inputs. Parallel directions have $\theta=0$ or $\pi$, so $\sin\theta=0$ and the cross product is zero. Conversely $\mathbf a\times\mathbf b=\mathbf0$ gives $|\mathbf a||\mathbf b|\sin\theta=0$; nonzero lengths force $\sin\theta=0$, hence $\theta=0$ or $\pi$. In particular $\mathbf a\times\mathbf a=\mathbf0$.
3. Perpendicular iff maximal magnitude, for nonzero inputs. If $\theta=\pi/2$, then $|\mathbf a\times\mathbf b|=|\mathbf a||\mathbf b|\sin(\pi/2)=|\mathbf a||\mathbf b|$. Conversely that equality gives $\sin\theta=1$, so $\theta=\pi/2$.
4. Anti-commutativity. The right-hand rule changes $\widehat{\mathbf n}$ to $-\widehat{\mathbf n}$ when the inputs are swapped:

$$\mathbf b\times\mathbf a=|\mathbf b||\mathbf a|\sin\theta(-\widehat{\mathbf n})=-|\mathbf a||\mathbf b|\sin\theta\widehat{\mathbf n}=-\mathbf a\times\mathbf b.$$

Thus $\mathbf a\times\mathbf b=-(\mathbf b\times\mathbf a)$.

5. Compatibility with scalars. $k(\mathbf a\times\mathbf b)=(k\mathbf a)\times\mathbf b=\mathbf a\times(k\mathbf b)$. The source’s magnitude proof reads

$$\begin{aligned} k(\mathbf a\times\mathbf b)&=k|\mathbf a||\mathbf b|\sin\theta\widehat{\mathbf n}\\ &=|k\mathbf a||\mathbf b|\sin\theta\widehat{\mathbf n}=(k\mathbf a)\times\mathbf b\\ &=|\mathbf a||k\mathbf b|\sin\theta\widehat{\mathbf n}=\mathbf a\times(k\mathbf b). \end{aligned}$$

That same-normal magnitude chain requires positive k; for negative k, the angle and normal direction also change. The algebraic identity is still valid for every real k, including zero, as the Cartesian formula confirms. The source calls this an associative law; it concerns a scalar factor, not arbitrary regrouping of three vector cross products.

6. Distributivity, in both orders:

$$\mathbf a\times(\mathbf b+\mathbf c)=\mathbf a\times\mathbf b+\mathbf a\times\mathbf c,\qquad (\mathbf b+\mathbf c)\times\mathbf a=\mathbf b\times\mathbf a+\mathbf c\times\mathbf a.$$

Example 16 combines anti-commutativity and distributivity into a zero cross product. Check that both factors are nonzero before using its geometric parallelism conclusion.