Q. 4 - Calculus

Question

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

Step-by-Step Solution

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 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 = ( u 1 , u 2 , u 3 )   and   v = ( v 1 , v 2 , v 3 ) .

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

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

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