Cross Product

Determinants

Before learning about cross product, it is useful to review some basic information about determinants:

In two dimensions

(Note: According to some texts, the operation of cross product is not actually defined in two dimensions, it is only defined in three dimensions. The 2d vector operation described here is often called a perp dot product for reasons described below.)

The cross product of two 2-dimensional vectors \(\mathbf{u} = \left\langle u_1, u_2\right\rangle\) and \(\mathbf{v} = \left\langle v_1, v_2\right\rangle\) is defined as the scalar

\[\mathbf{u}\times\mathbf{v} = \left\lvert\mathbf{u} \right\rvert\left\lvert\mathbf{v} \right\rvert\sin\theta\]

where \(\theta\) is the angle between \(\mathbf{u}\) and \(\mathbf{v}\).

It is equal to the “signed area” of the parallelogram defined by the two vectors:

(The area of a parallelogram is equal to the base times height. In the parallelogram above, the base is \(\left\lvert\mathbf{u} \right\rvert\) and the height is \(\left\lvert\mathbf{v} \right\rvert\sin\theta\). If \(\theta <0\), we get a “negative” area”.)

In components, it turns out the cross product or vectors \(\mathbf{u} = \left\langle u_1, u_2\right\rangle\) and \(\mathbf{v} = \left\langle v_1, v_2\right\rangle\) is

\[\mathbf{u}\times\mathbf{v} = u_1v_2 - v_1u_2\]

Explanation and connection to dot product:

The following illustration shows two vectors: \(\mathbf{u}\) and \(\mathbf{v}\), together with a vector \(\mathbf{u}^\perp\), which is a vector of the same magnitude as \(\mathbf{u}\), but perpendicular to \(\mathbf{u}\). If the component form of \(\mathbf{u}\) is \(\left\langle u_1,u_2\right\rangle\), the comoponent form of \(\mathbf{u}^\perp\) is \(\left\langle -u_2, u_1\right\rangle\) (You can also use \(\left\langle u_2, -u_1\right\rangle\), which would point in the opposite direction. You can verify that you will get the same result in the end.)

Then

\[\begin{aligned} \mathbf{u}^\perp \cdot \mathbf{v} &= \left\lvert\mathbf{u}^\perp \right\rvert\left\lvert\mathbf{v} \right\rvert\cos\left(\frac{\pi}{2}-\theta\right)\\ &=\left\lvert\mathbf{u} \right\rvert\left\lvert\mathbf{v} \right\rvert\sin\left(\theta\right)\\ &=\mathbf{u}\times\mathbf{v} \end{aligned}\]

In components, \(\mathbf{u}^\perp \cdot \mathbf{v} = -u_2v_1 + u_1v_2 = u_1v_2 - u_2v_1\).

This is why the “2d cross product” is sometimes called a perp dot product. The notation is then \(\mathbf{u}\perp\mathbf{v}\). Another term used for this operation is the outer product.

Properties

From the definition and the equivalent component formula, we get the following properties:

Anti-commutativity: \(\mathbf{u}\times\mathbf{v} = -\mathbf{v}\times\mathbf{u}\)
Nilpotent property: \(\mathbf{u}\times\mathbf{u} = 0\)
Distributive property: \(\mathbf{u}\times\left(\mathbf{v} + \mathbf{w}\right) = (\mathbf{u}\times\mathbf{v}) + (\mathbf{u}\times\mathbf{w})\)
Associativity with scalar multiple: \(c(\mathbf{u}\times\mathbf{v} = (c\mathbf{u})\times\mathbf{v} = \mathbf{u}\times(c\mathbf{v})\)
Lagrange property: \((\mathbf{u}\times\mathbf{v})^2 + (\mathbf{u}\cdot\mathbf{v})^2 = (\left\lvert u \right\rvert\left\lvert v \right\rvert)^2\)

(The last one follows directly from the Pythagorean identity for sine and cosine.)

Calculating using determinants

The 2d cross product \(\mathbf{u}\times\mathbf{v}\) can also be defined as a determinant of a matrix with first row \(\mathbf{u}\) and second row \(\mathbf{v}\):

\[\mathbf{u}\times\mathbf{v} = \left\lvert \begin{matrix} u_1 & u_2\\ v_1 & v_2 \end{matrix} \right\rvert = u_1v_2 - u_2v_1\]

Example

Find the area of the parallelogram defined by the vectors \(\mathbf{a} = \left\langle 3,2\right\rangle\) and \(\mathbf{b} = \left\langle 1,-5\right\rangle\).

The signed area of the parallelogram is

\[\mathbf{a}\times \mathbf{b} = \left\lvert\begin{matrix}3 & 2\\1 & -5\end{matrix}\right\rvert = 3\cdot (-5) - 2\cdot 1 = -17\]

so the area is 17 square units.

Cross product in 3 dimensions

The true cross product of two vectors is defined when the two vectors are 3d vectors. The cross product of two 3d vectors \(\mathbf{u}\) and \(\mathbf{v}\) (also known as the vector product or outer product) is a vector with the following properties:

It is perpendicular to both \(\mathbf{u}\) and \(\mathbf{v}\).
Its magnitude is equal to the area of the parallelogram defined by the vectors \(\mathbf{u}\) and \(\mathbf{v}\).
The direction is determined by the right hand rule.

In components, the cross product can be calculated using the following determinant:

\[\begin{aligned} \mathbf{u}\times\mathbf{v} = \left\lvert\begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k}\\ u_1 & u_2 & u_3\\ v_1 & v_2 & v_3 \end{matrix}\right\rvert &= \left\langle \left\lvert\begin{matrix}u_2 & u_3\\v_2 & v_3\end{matrix}\right\rvert, -\left\lvert\begin{matrix}u_1 & u_3\\v_1 & v_3\end{matrix}\right\rvert, \left\lvert\begin{matrix}u_1 & u_2\\v_1 & v_2\end{matrix}\right\rvert\right\rangle\\ &= \left\langle u_2v_3 - u_3v_2, -(u_1v_3 - u_3v_1), u_1v_2 - u_2v_1\right\rangle \end{aligned}\]

Try it yourself

Properties

Anti-commutativity: \(\mathbf{u}\times\mathbf{v} = -\mathbf{v}\times\mathbf{u}\)
Nilpotent property: \(\mathbf{u}\times\mathbf{u} = \left\langle 0,0,0\right\rangle = \mathbf{O}\)
Zero vector product: \(\mathbf{O}\times\mathbf{u} = \mathbf{u}\times\mathbf{O} = \mathbf{O}\)
Distributive property: \(\mathbf{u}\times\left(\mathbf{v} + \mathbf{w}\right) = (\mathbf{u}\times\mathbf{v}) + (\mathbf{u}\times\mathbf{w})\)
Associativity with scalar multiple: \(c(\mathbf{u}\times\mathbf{v} = (c\mathbf{u})\times\mathbf{v} = \mathbf{u}\times(c\mathbf{v})\)
Lagrange property: \(\left\lvert\mathbf{u}\times\mathbf{v} \right\rvert^2 + (\mathbf{u}\cdot\mathbf{v})^2 = (\left\lvert u \right\rvert\left\lvert v \right\rvert)^t2\)
Orthogonality: \((\mathbf{u}\times\mathbf{v})\cdot\mathbf{u} = (\mathbf{u}\times\mathbf{v})\cdot\mathbf{v} = 0\)
Triple scalar product: \(\mathbf{u}\cdot(\mathbf{v}\times\mathbf{w}) = (\mathbf{u}\times\mathbf{v})\cdot\mathbf{w}\)

The first three properties follow directly from the properties of determinants:

Switching two rows in a determinant changes the sign of the determinant
Determinant with two equal rows is 0
Determinant with a row of zeros is 0

The Lagrange property is a consequence of the Pythagorean identity for sine and cosine. The rest of the properties can be proven by working with components.

For us, the most useful properties will be the orthogonality and the fact that \(\left\lvert\mathbf{u}\times\mathbf{v} \right\rvert\) is the area of the parallelogram defined by \(\mathbf{u}\) and \(\mathbf{v}\).

Try it yourself

Triple Scalar Product

The triple scalar product \(\mathbf{u}\cdot(\mathbf{v}\times\mathbf{w})\) is related to volume on the following way:

It also gives another interpretation to a \(3\times 3\) determinant. The determinant

\[\left\lvert\begin{matrix} a_1 & a_2 & a_3\\ b_1 & b_2 & b_3\\ c_1 & c_2 & c_3\\ \end{matrix}\right\rvert\]

is the signed volume of the parallelepiped determined by the vectors \(\mathbf{a} = \left\langle a_1, a_2, a_3\right\rangle\), \(\mathbf{b} = \left\langle b_1, b_2, b_3\right\rangle\), and \(\mathbf{c} = \left\langle c_1, c_2, c_3\right\rangle\), where the sign is determined by the order in which the vectors are listed.