Answer to Question #107895 in Analytic Geometry for Caylin

Question #107895
Suppose u, v and w are vectors in 3-space,where u =(u1u2, u3), v =(v1, v2, v3) and w= (w1, w2, w3).

Express ( u * v) * w as a determinant.
1
Expert's answer
2020-04-03T16:58:28-0400

"u=(u_1,u_2,u_3), v=(v_1,v_2,v_3), w=(w_1,w_2,w_3)"

"u\\times v=\\begin{vmatrix}\n \\vec{i} &\\vec{j}&\\vec{k} \\\\\n u_1 &u_2&u_3\\\\\nv_1&v_2&v_3\n\\end{vmatrix}=\\\\\n=\\vec{i}\\cdot(u_2v_3-u_3v_2)-\\vec{j}\\cdot(u_1v_3-u_3v_1)+\\\\\n+\\vec{k}\\cdot(u_1v_2-u_2v_1)=\\\\\n=(u_2v_3-u_3v_2,-u_1v_3+u_3v_1,u_1v_2-u_2v_1)\\\\\n(u\\times v)\\times w=\\\\\n=\\begin{vmatrix}\n \\vec{i} &\\vec{j}&\\vec{k} \\\\\n u_2v_3-u_3v_2&-u_1v_3+u_3v_1&u_1v_2-u_2v_1\\\\\nw_1&w_2&w_3\n\\end{vmatrix}"



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