Q. 78 - Calculus

Question

Let $u=(u_1,u_2,u_3), v=(v_1,v_2,v_3), \mathrm{and} w=(w_1,w_2,w_3)$. Show that

$u·(v\times w)=\det \left[\begin{array}{ccc}{\mathrm{u}}_1& {\mathrm{u}}_2& {\mathrm{u}}_3\\ {\mathrm{v}}_1& {\mathrm{v}}_2& {\mathrm{v}}_3\\ {\mathrm{w}}_1& {\mathrm{w}}_2& {\mathrm{w}}_3\end{array}\right]$

Step-by-Step Solution

Answer

Hence, we prove that $u·(v\times w)=\det \left[\begin{array}{ccc}{\mathrm{u}}_1& {\mathrm{u}}_2& {\mathrm{u}}_3\\ {\mathrm{v}}_1& {\mathrm{v}}_2& {\mathrm{v}}_3\\ {\mathrm{w}}_1& {\mathrm{w}}_2& {\mathrm{w}}_3\end{array}\right]$

Step 1. Given Information

Let $u=(u_1,u_2,u_3), v=(v_1,v_2,v_3), \mathrm{and} w=(w_1,w_2,w_3)$. Show that

$u·(v\times w)=\det \left[\begin{array}{ccc}{\mathrm{u}}_1& {\mathrm{u}}_2& {\mathrm{u}}_3\\ {\mathrm{v}}_1& {\mathrm{v}}_2& {\mathrm{v}}_3\\ {\mathrm{w}}_1& {\mathrm{w}}_2& {\mathrm{w}}_3\end{array}\right]$

Step 2. Given that u = ( u 1 , u 2 , u 3 ) ,   v = ( v 1 , v 2 , v 3 ) ,   and   w = ( w 1 , w 2 , w 3 )

We firstly finding the value of $v\times w$

$$\begin{gathered} v\times w=\det \left[\begin{array}{ccc}i& j& k\\ {\mathrm{v}}_1& {\mathrm{v}}_2& {\mathrm{v}}_3\\ {\mathrm{w}}_1& {\mathrm{w}}_2& {\mathrm{w}}_3\end{array}\right] \\ v\times w=i\left|\begin{array}{cc}{\mathrm{v}}_2& {\mathrm{v}}_3\\ w_2& w_3\end{array}\right|-j\left|\begin{array}{cc}{\mathrm{v}}_1& {\mathrm{v}}_3\\ w_1& w_3\end{array}\right|+k\left|\begin{array}{cc}{\mathrm{v}}_1& {\mathrm{v}}_2\\ w_1& w_2\end{array}\right| \\ v\times w=i({\mathrm{v}}_2{\mathrm{w}}_3-{\mathrm{v}}_3{\mathrm{w}}_2)-\mathrm{j}({\mathrm{v}}_1{\mathrm{w}}_3-{\mathrm{v}}_3{\mathrm{w}}_1)+\mathrm{k}({\mathrm{v}}_1{\mathrm{w}}_2-{\mathrm{v}}_2{\mathrm{w}}_1) \\ \mathrm{v}\times \mathrm{w}=({\mathrm{v}}_2{\mathrm{w}}_3-{\mathrm{v}}_3{\mathrm{w}}_2,{\mathrm{v}}_1{\mathrm{w}}_3-{\mathrm{v}}_3{\mathrm{w}}_1,{\mathrm{v}}_1{\mathrm{w}}_2-{\mathrm{v}}_2{\mathrm{w}}_1) \end{gathered}$$

Step 3. Now finding the value of u · ( v × w )

$$\begin{gathered} u·(v\times w)=(u_1,u_2,u_3)·({\mathrm{v}}_2{\mathrm{w}}_3-{\mathrm{v}}_3{\mathrm{w}}_2,{\mathrm{v}}_1{\mathrm{w}}_3-{\mathrm{v}}_3{\mathrm{w}}_1,{\mathrm{v}}_1{\mathrm{w}}_2-{\mathrm{v}}_2{\mathrm{w}}_1) \\ u·(v\times w)=u_1({\mathrm{v}}_2{\mathrm{w}}_3-{\mathrm{v}}_3{\mathrm{w}}_2)+{\mathrm{u}}_2({\mathrm{v}}_1{\mathrm{w}}_3-{\mathrm{v}}_3{\mathrm{w}}_1)+{\mathrm{u}}_3({\mathrm{v}}_1{\mathrm{w}}_2-{\mathrm{v}}_2{\mathrm{w}}_1) \\ u·(v\times w)=u_1{\mathrm{v}}_2{\mathrm{w}}_3-u_1{\mathrm{v}}_3{\mathrm{w}}_2+{\mathrm{u}}_2{\mathrm{v}}_1{\mathrm{w}}_3-{\mathrm{u}}_2{\mathrm{v}}_3{\mathrm{w}}_1+{\mathrm{u}}_3{\mathrm{v}}_1{\mathrm{w}}_2-{\mathrm{u}}_3{\mathrm{v}}_2{\mathrm{w}}_1 \end{gathered}$$

Step 4. Now solving the det u 1 u 2 u 3 v 1 v 2 v 3 w 1 w 2 w 3

$$\begin{gathered} \det \left[\begin{array}{ccc}{\mathrm{u}}_1& {\mathrm{u}}_2& {\mathrm{u}}_3\\ {\mathrm{v}}_1& {\mathrm{v}}_2& {\mathrm{v}}_3\\ {\mathrm{w}}_1& {\mathrm{w}}_2& {\mathrm{w}}_3\end{array}\right]={\mathrm{u}}_1\left|\begin{array}{cc}{\mathrm{v}}_2& {\mathrm{v}}_3\\ {\mathrm{w}}_2& {\mathrm{w}}_3\end{array}\right|-{\mathrm{u}}_2\left|\begin{array}{cc}{\mathrm{v}}_1& {\mathrm{v}}_3\\ {\mathrm{w}}_1& {\mathrm{w}}_3\end{array}\right|+{\mathrm{u}}_3\left|\begin{array}{cc}{\mathrm{v}}_1& {\mathrm{v}}_2\\ {\mathrm{w}}_1& {\mathrm{w}}_2\end{array}\right| \\ \det \left[\begin{array}{ccc}{\mathrm{u}}_1& {\mathrm{u}}_2& {\mathrm{u}}_3\\ {\mathrm{v}}_1& {\mathrm{v}}_2& {\mathrm{v}}_3\\ {\mathrm{w}}_1& {\mathrm{w}}_2& {\mathrm{w}}_3\end{array}\right]={\mathrm{u}}_1({\mathrm{v}}_2{\mathrm{w}}_3-{\mathrm{v}}_3{\mathrm{w}}_2)-{\mathrm{u}}_2({\mathrm{v}}_1{\mathrm{w}}_3-{\mathrm{v}}_3{\mathrm{w}}_1)+{\mathrm{u}}_3({\mathrm{v}}_1{\mathrm{w}}_2-{\mathrm{v}}_2{\mathrm{w}}_1) \\ \det \left[\begin{array}{ccc}{\mathrm{u}}_1& {\mathrm{u}}_2& {\mathrm{u}}_3\\ {\mathrm{v}}_1& {\mathrm{v}}_2& {\mathrm{v}}_3\\ {\mathrm{w}}_1& {\mathrm{w}}_2& {\mathrm{w}}_3\end{array}\right]={\mathrm{u}}_1{\mathrm{v}}_2{\mathrm{w}}_3-{\mathrm{u}}_1{\mathrm{v}}_3{\mathrm{w}}_2-{\mathrm{u}}_2{\mathrm{v}}_1{\mathrm{w}}_3-{\mathrm{u}}_2{\mathrm{v}}_3{\mathrm{w}}_1+{\mathrm{u}}_3{\mathrm{v}}_1{\mathrm{w}}_2-{\mathrm{u}}_3{\mathrm{v}}_2{\mathrm{w}}_1 \end{gathered}$$

Hence, $u·(v\times w)=\det \left[\begin{array}{ccc}{\mathrm{u}}_1& {\mathrm{u}}_2& {\mathrm{u}}_3\\ {\mathrm{v}}_1& {\mathrm{v}}_2& {\mathrm{v}}_3\\ {\mathrm{w}}_1& {\mathrm{w}}_2& {\mathrm{w}}_3\end{array}\right]$