Question

For the Cartesian form of the distance between skew lines, which determinant is the numerator?

Options

• \[\begin{vmatrix}\mathbf a_1&\mathbf b_1&\mathbf c_1\\\mathbf a_2&\mathbf b_2&\mathbf c_2\\x_1-x_2&y_1-y_2&z_1-z_2\end{vmatrix}\]

• \[\begin{vmatrix}x_2-x_1&y_2-y_1&z_2-z_1\\\mathbf a_1&\mathbf b_1&\mathbf c_1\\\mathbf a_2&\mathbf b_2&\mathbf c_2\end{vmatrix}\]

• \[\begin{vmatrix}x_1&y_1&z_1\\x_2&y_2&z_2\\\mathbf a_1&\mathbf b_1&\mathbf c_1\end{vmatrix}\]

• \[\begin{vmatrix}\mathbf a_1&\mathbf a_2&x_2-x_1\\\mathbf b_1&\mathbf b_2&y_2-y_1\\\mathbf c_1&\mathbf c_2&z_2-z_1\end{vmatrix}\]

Answer 1

The Cartesian numerator uses the coordinate differences \[x_2-x_1\], \[y_2-y_1\], and \[z_2-z_1\] in its first row. The next two rows contain the components of the two direction vectors.