Q. 66 - Calculus

Question

Use the definition of the cross product to prove that the cross product of two vectors u and v is anti-commutative; that is, prove that $u\times v=-v\times u$. (This is

Theorem 10.27.)

Step-by-Step Solution

Answer

Hence, we prove that $u\times v=-(v\times u)$.

Step 1. Given Information

Use the definition of the cross product to prove that the cross product of two vectors u and v is anti-commutative; that is, prove that $u\times v=-v\times u$. (This is Theorem 10.27.)

Step 2. Theorem 10.27

Theorem states that "For any vectors u and v in ${\mathrm{ℝ}}^3$ $u\times v=-(v\times u)$."

Step 3. We have to prove u × v = − ( v × u )

Firstly finding the value of $u\times v$

$$\begin{gathered} u\times v=(u_1,u_2,u_3)\times (v_1,v_2,v_3) \\ u\times v=\left[\begin{array}{ccc}i& j& k\\ u_1& u_2& u_3\\ v_1& v_2& v_3\end{array}\right] \\ u\times v=i\left|\begin{array}{cc}{u}_2& u_3\\ v_2& v_3\end{array}\right|-j\left|\begin{array}{cc}{u}_1& u_3\\ v_1& v_3\end{array}\right|+k\left|\begin{array}{cc}{u}_1& u_2\\ v_1& v_2\end{array}\right| \\ u\times v=i(u_2{v}_3-u_3{v}_2)-j(u_1{v}_3-u_3{v}_1)+k(u_1{v}_2-u_2{v}_1) \\ u\times v=(u_2{v}_3-u_3{v}_2,-u_1{v}_3+u_3{v}_1,u_1{v}_2-u_2{v}_1) \end{gathered}$$

Step 4. Now finding the value of v × u

$$\begin{gathered} v\times u=(u_1,u_2,u_3)\times (v_1,v_2,v_3) \\ v\times u=\left[\begin{array}{ccc}i& j& k\\ v_1& v_2& v_3\\ u_1& u_2& u_3\end{array}\right] \\ v\times u=i\left|\begin{array}{cc}{v}_2& v_3\\ u_2& u_3\end{array}\right|-j\left|\begin{array}{cc}{v}_1& v_3\\ u_1& u_3\end{array}\right|+k\left|\begin{array}{cc}{v}_1& v_2\\ u_1& u_2\end{array}\right| \\ v\times u=i(v_2{u}_3-v_3{u}_2)-j(v_1{u}_3-v_3{u}_1)+k(v_1{u}_2-v_2{u}_1) \\ v\times u=(v_2{u}_3-v_3{u}_2,-v_1{u}_3+v_3{u}_1,v_1{u}_2-v_2{u}_1) \\ v\times u=-(u_3{v}_2-u_2{v}_3,-u_3{v}_1+u_1{v}_3,u_2{v}_1-u_1{v}_2) \end{gathered}$$

Hence it shows that $u\times v=-(v\times u)$