# Paper Homework 2

Problem 2: Most mistakes come from the miscalculation of the cross product. I strongly recommend that students should memorize by determinant as follows. $$\langle a_1, b_1, c_1\rangle\times \langle a_2, b_2, c_2\rangle=\det\begin{pmatrix}\vec{i} & \vec{j} & \vec{k}\\ a_1 & b_1 & c_1\\ a_2 & b_2 & c_2\end{pmatrix}$$

Problem 6: The key point to solve this problem is to observe the distance between two skewed lines is equal to the projection of any vector between two points on the lines onto the normal vector determined by the directions of the lines.

Problem 8: You need to subtract constants to balance the constants you added to complete the squares. Few students got the constant wrong hence land on the wrong classification of the surface. Since the negative sign is on the $y$ term, the axis of the surface is parallel to the $y$-axis.

Problem 10: Please pay attention to the change of sings when subtracting two equations.