Answer to Question #114831 in Analytic Geometry for Reginald Thebe

Question #114831
Suppose u and v are nonzero vectors in 3-space, where u=(u1,u2, u3) and v=(v1,v2,v3). Prove that u × v is perpendicular to both u and v by making use of the dot product
1
Expert's answer
2020-05-08T19:45:07-0400

We have u×v=u1u2u3v1v2v3e1e2e3=u\times v=\begin{vmatrix} u_1&u_2&u_3\\ v_1&v_2&v_3\\ e_1&e_2&e_3 \end{vmatrix}=

=(u2u3v2v3,u1u3v1v3,u1u2v1v2)=\left(\begin{vmatrix} u_2&u_3\\ v_2&v_3\\ \end{vmatrix},-\begin{vmatrix} u_1&u_3\\ v_1&v_3\\ \end{vmatrix},\begin{vmatrix} u_1&u_2\\ v_1&v_2\\ \end{vmatrix}\right)

Then (u,u×v)=u1u2u3v2v3u2u1u3v1v3+u3u1u2v1v2=(u,u\times v)=u_1\begin{vmatrix} u_2&u_3\\ v_2&v_3\\ \end{vmatrix}-u_2\begin{vmatrix} u_1&u_3\\ v_1&v_3\\ \end{vmatrix}+u_3\begin{vmatrix} u_1&u_2\\ v_1&v_2\\ \end{vmatrix}=

=u1u2u3v1v2v3u1u2u3=0=\begin{vmatrix} u_1&u_2&u_3\\ v_1&v_2&v_3\\ u_1&u_2&u_3 \end{vmatrix}=0 and (v,u×v)=v1u2u3v2v3v2u1u3v1v3+v3u1u2v1v2=(v,u\times v)=v_1\begin{vmatrix} u_2&u_3\\ v_2&v_3\\ \end{vmatrix}-v_2\begin{vmatrix} u_1&u_3\\ v_1&v_3\\ \end{vmatrix}+v_3\begin{vmatrix} u_1&u_2\\ v_1&v_2\\ \end{vmatrix}=

=u1u2u3v1v2v3v1v2v3=0=\begin{vmatrix} u_1&u_2&u_3\\ v_1&v_2&v_3\\ v_1&v_2&v_3 \end{vmatrix}=0


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