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For $$\vec{a} = a_x \hat{i} + a_y \hat{j} + a_z \hat{k}$$ and $$\vec{b} = b_x \hat{i} + b_y \hat{j} + b_z \hat{k}$$:
$$\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix}$$
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For $$\vec{a} = a_x \hat{i} + a_y \hat{j} + a_z \hat{k}$$ and $$\vec{b} = b_x \hat{i} + b_y \hat{j} + b_z \hat{k}$$:
$$\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix}$$
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