Question

How is the determinant of a 3 × 3 matrix used in the computation of the determinant of two vectors$u=(u_1,u_2,u_3)\mathrm{and}v=(v_1,v_2,v_3)$?

Short answer

The determinant of a 3 × 3 matrix is $u\times v=\det \left[\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ {\mathrm{u}}_1& {\mathrm{u}}_2& {\mathrm{u}}_3\\ {\mathrm{v}}_1& {\mathrm{v}}_2& {\mathrm{v}}_3\end{array}\right]$.

Step-by-step solution

Step 1. Given Information 

We have to find how the determinant of a 3 × 3 matrix used in the computation of the determinant of two vectors$u=(u_1,u_2,u_3)\mathrm{and}v=(v_1,v_2,v_3)$.

Step 2. The given two vectors are u=(u1,u2,u3) and v=(v1,v2,v3).

$$\begin{gathered} u=(u_1,u_2,u_3)v=(v_1,v_2,v_3) \\ u=u_{1}i+u_{2}j+u_{3}kv=v_{1}i+v_{2}j+v_{3}k \end{gathered}$$

Step 3. Now the determinant of a 3 × 3 matrix is

$u\times v=\det \left[\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ {\mathrm{u}}_1& {\mathrm{u}}_2& {\mathrm{u}}_3\\ {\mathrm{v}}_1& {\mathrm{v}}_2& {\mathrm{v}}_3\end{array}\right]$