# Math Insight

### The relationship between determinants and area or volume

From the properties of the geometric definition of the cross product and the scalar triple product, we can discover a link between $2 \times 2$ determinants and area, and a link between $3 \times 3$ determinants and volume.

#### 2 $\times$ 2 determinants and area

The area of the parallelogram spanned by $\vc{a}$ and $\vc{b}$ is the magnitude of $\vc{a} \times \vc{b}$. We can write the cross product of $\vc{a} = a_1\vc{i}+a_2\vc{j}+a_3\vc{k}$ and $\vc{b} = b_1\vc{i}+b_2\vc{j}+b_3\vc{k}$ as the determinant \begin{align*} \vc{a} \times \vc{b}= \left| \begin{array}{ccc} \vc{i} & \vc{j} & \vc{k}\\ a_1 & a_2 & a_3\\ b_1 & b_2 & b_3 \end{array} \right|. \end{align*}

Now, imagine that $\vc{a}$ and $\vc{b}$ lie in the plane so that $a_3 = b_3=0$. Using the rules for calculating determinants, we see that, in this case, the cross product simplifies to \begin{align*} \vc{a} \times \vc{b}= \left| \begin{array}{cc} a_1 & a_2\\ b_1 & b_2 \end{array} \right| \vc{k}. \end{align*}

Hence, the area of the parallelogram, $\| \vc{a} \times \vc{b}\|$, is the absolute value of the determinant \begin{align*} \left| \begin{array}{cc} a_1 & a_2\\ b_1 & b_2 \end{array} \right|. \end{align*} As mentioned in the discussion about determinant notation, it's difficult to represent the absolute value of a determinant using the above notation. Instead, we write that the area of the parallelogram spanned by $\vc{a} = a_1 \vc{i} + a_2 \vc{j}$ and $\vc{b} = b_1 \vc{i} + b_2\vc{j}$ is \begin{align*} \| \vc{a} \times \vc{b}\| =\left|\det \left(\left[ \begin{array}{cc} a_1 & a_2\\ b_1 & b_2 \end{array} \right]\right) \right|. \end{align*}

#### 3 $\times$ 3 determinants and volume

The volume of a parallelepiped spanned by the vectors $\vc{a}$, $\vc{b}$ and $\vc{c}$ is the absolute value of the scalar triple product $(\vc{a} \times \vc{b}) \cdot \vc{c}$. We can write the scalar triple product of $\vc{a} = a_1\vc{i}+a_2\vc{j}+a_3\vc{k}$, $\vc{b} = b_1\vc{i}+b_2\vc{j}+b_3\vc{k}$, and $\vc{c} = c_1\vc{i}+c_2\vc{j}+c_3\vc{k}$ as the determinant \begin{align*} (\vc{a} \times \vc{b}) \cdot \vc{c} = \left| \begin{array}{ccc} c_1 & c_2 & c_3\\ a_1 & a_2 & a_3\\ b_1 & b_2 & b_3 \end{array} \right|. \end{align*} Hence, using the alternative notation for determinant, the volume of the parallelepiped spanned by $\vc{a}$, $\vc{b}$, and $\vc{c}$ is \begin{align*} |(\vc{a} \times \vc{b}) \cdot \vc{c}| = \left|\det \left(\left[ \begin{array}{ccc} c_1 & c_2 & c_3\\ a_1 & a_2 & a_3\\ b_1 & b_2 & b_3 \end{array} \right]\right) \right|. \end{align*}