Question

Find A X B when A = 2i-3j-k and B = i + 4j -2k

Answer 1

By the definition of the cross product we have:

$$\vec{a}\times \vec{b}=\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix},$$ $$\vec{a}\times \vec{b}=\hat{i}\begin{pmatrix} a_y & a_z \\ b_y & b_z \end{pmatrix} - \hat{j}\begin{pmatrix} a_x & a_z \\ b_x & b_z \end{pmatrix} +\hat{k}\begin{pmatrix} a_x & a_y \\ b_x & b_y \end{pmatrix},$$ $$\vec{a}\times \vec{b}=(a_yb_z-a_zb_y)\hat{i}-(a_xb_z-a_zb_x)\hat{j}+(a_xb_y-a_yb_x)\hat{k},$$ $$\vec{a}\times \vec{b}=(6-(-4))\hat{i}-(-4-(-1))\hat{j}+(8-(-3))\hat{k},$$ $$\vec{a}\times \vec{b}=10\hat{i}+3\hat{j}+11\hat{k}.$$