The Cross Product

Definition For vectors $\mathbf{a} = \langle a_1, a_2, a_3 \rangle$ and $\mathbf{b} = \langle b_1, b_2, b_3 \rangle$ in three-dimensional space, the cross product is the vector whose components are listed below as a column, first component on top,

The column is only a narrower way to write the same vector. In angle brackets it is $\langle a_2 b_3 - a_3 b_2, \ a_3 b_1 - a_1 b_3, \ a_1 b_2 - a_2 b_1 \rangle$. A convenient way to remember it is the symbolic determinant

expanded along the first row with the middle term subtracted,

It is only a memory aid, since a real determinant has numbers in every entry.

Definition For all vectors $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ in space and every real number $t$,

The cross product is not commutative, and it is not associative. In general $(\mathbf{a} \times \mathbf{b}) \times \mathbf{c} \ne \mathbf{a} \times (\mathbf{b} \times \mathbf{c})$.

Definition Let $\mathbf{a}$ and $\mathbf{b}$ be nonzero, nonparallel vectors with angle $\theta$, where $0 < \theta < \pi$. In right-handed coordinates, $\mathbf{a} \times \mathbf{b}$ is the unique vector that

1. is perpendicular to both $\mathbf{a}$ and $\mathbf{b}$,
2. has length $\lvert \mathbf{a} \rvert \, \lvert \mathbf{b} \rvert \sin \theta$,
3. points the way your right thumb points when the fingers curl from $\mathbf{a}$ toward $\mathbf{b}$ through the angle $\theta$.

The length formula also holds for nonzero parallel vectors, where $\theta = 0$ or $\theta = \pi$ and both sides are $0$. Since $\sin \theta > 0$ for $0 < \theta < \pi$, two nonzero vectors are parallel exactly when $\mathbf{a} \times \mathbf{b} = \mathbf{0}$. If $\mathbf{a}$ or $\mathbf{b}$ is $\mathbf{0}$, the cross product is $\mathbf{0}$ and no angle is needed. For the basis vectors,

Definition The parallelogram with adjacent sides $\mathbf{a}$ and $\mathbf{b}$ has area

The triangle with sides $\mathbf{a}$ and $\mathbf{b}$ from a common vertex has half this area. If $\mathbf{a}$ and $\mathbf{b}$ are not parallel, $\mathbf{a} \times \mathbf{b}$ is a nonzero normal vector to every plane that contains both directions.

Definition If a force $\mathbf{F}$ acts at a point whose position relative to a pivot $O$ is $\mathbf{r}$, the torque about $O$ is

where $\theta$ is the angle between $\mathbf{r}$ and $\mathbf{F}$. When the torque is nonzero it points along the axis about which the force tends to rotate the object, and by the right-hand rule the rotation looks counterclockwise from the tip of $\boldsymbol{\tau}$. With meters and newtons, torque is in newton meters.

Definition The scalar triple product of $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ is the number $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})$. It equals the determinant

It is unchanged by a cyclic shift of the three vectors and changes sign when any two of them are swapped, so

Definition The parallelepiped with edges $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ from one vertex has volume

The triple product itself is a signed volume. It is zero exactly when the three vectors are coplanar, that is, when they lie in one plane once they are drawn from a common initial point. Three vectors that are not coplanar form a right-handed ordered triple $(\mathbf{a}, \mathbf{b}, \mathbf{c})$ when $\mathbf{c}$ lies on the side of the plane of $\mathbf{a}$ and $\mathbf{b}$ that your right thumb points to as your fingers curl from $\mathbf{a}$ toward $\mathbf{b}$, which is the side $\mathbf{a} \times \mathbf{b}$ points to. Otherwise the triple is left-handed. In right-handed coordinates the identity $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c}$ shows that the triple product is positive for a right-handed triple and negative for a left-handed one. For example $\mathbf{i} \cdot (\mathbf{j} \times \mathbf{k}) = 1$. The tetrahedron with the same three edges has volume $\frac{1}{6}V$.