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

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

Short 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-by-step solution

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

role="math" localid="1650039947035" $u·(v\times w)=\det \left[\begin{array}{ccc}{\mathrm{u}}_1& {\mathrm{u}}_2& {\mathrm{u}}_3\\ {\mathrm{v}}_1& {\mathrm{v}}_2& {\mathrm{v}}_3\\ {\mathrm{w}}_1& {\mathrm{w}}_2& {\mathrm{w}}_3\end{array}\right]$

Step 2. Given that u=(u1,u2,u3), v=(v1,v2,v3), and w=(w1,w2,w3)

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

role="math" localid="1650039221614" $$\begin{gathered} v\times w=\det \left[\begin{array}{ccc}i& j& k\\ {\mathrm{v}}_1& {\mathrm{v}}_2& {\mathrm{v}}_3\\ {\mathrm{w}}_1& {\mathrm{w}}_2& {\mathrm{w}}_3\end{array}\right] \\ v\times w=i\left|\begin{array}{cc}{\mathrm{v}}_2& {\mathrm{v}}_3\\ w_2& w_3\end{array}\right|-j\left|\begin{array}{cc}{\mathrm{v}}_1& {\mathrm{v}}_3\\ w_1& w_3\end{array}\right|+k\left|\begin{array}{cc}{\mathrm{v}}_1& {\mathrm{v}}_2\\ w_1& w_2\end{array}\right| \\ v\times w=i({\mathrm{v}}_2{\mathrm{w}}_3-{\mathrm{v}}_3{\mathrm{w}}_2)-\mathrm{j}({\mathrm{v}}_1{\mathrm{w}}_3-{\mathrm{v}}_3{\mathrm{w}}_1)+\mathrm{k}({\mathrm{v}}_1{\mathrm{w}}_2-{\mathrm{v}}_2{\mathrm{w}}_1) \\ \mathrm{v}\times \mathrm{w}=({\mathrm{v}}_2{\mathrm{w}}_3-{\mathrm{v}}_3{\mathrm{w}}_2,{\mathrm{v}}_1{\mathrm{w}}_3-{\mathrm{v}}_3{\mathrm{w}}_1,{\mathrm{v}}_1{\mathrm{w}}_2-{\mathrm{v}}_2{\mathrm{w}}_1) \end{gathered}$$

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 detu1u2u3v1v2v3w1w2w3

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

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