Question

Which vector form gives the distance \[\mathbf d\] between skew lines?

Options

• \[\mathbf d=\left|\frac{(\overline{\mathbf b}_1\times\overline{\mathbf b}_2)\cdot(\overline{\mathbf a}_2-\overline{\mathbf a}_1)}{|\overline{\mathbf b}_1\times\overline{\mathbf b}_2|}\right|\]

• \[\mathbf d=\left|\frac{(\overline{\mathbf a}_2-\overline{\mathbf a}_1)\times\overline{\mathbf b}_1}{|\overline{\mathbf b}_1|}\right|\]

• \[\mathbf d=\left|\frac{(\overline{\mathbf a}_2-\overline{\mathbf a}_1)\cdot\overline{\mathbf b}_1}{|\overline{\mathbf b}_1|}\right|\]

• \[\mathbf d=\left|\frac{\overline{\mathbf b}_1\cdot\overline{\mathbf b}_2}{|\overline{\mathbf a}_2-\overline{\mathbf a}_1|}\right|\]

Answer 1

For skew lines, \[\overline{\mathbf b}_1\times\overline{\mathbf b}_2\] is perpendicular to both direction vectors. Its scalar product with \[\overline{\mathbf a}_2-\overline{\mathbf a}_1\], divided by its magnitude, gives the distance.