Past exam of the mathematics course of the University of Cambridge / 2008 / ia / Paper 1 / 5B (Vectors and Matrices)

In suffix notation, repeated indices are summed from one to three. The Levi-Civita symbol and Kronecker delta contraction isFor distinct free index pairs this follows from the two possible matching permutations and their signs; if either pair repeats an index both sides vanish. ThereforeRecognizing the dot products proves the vector triple product identity:

Use the Levi-Civita symbol contraction ${ϵ}_{ijk}{ϵ}_{i\ell m}={\delta }_{j\ell }{\delta }_{km}-{\delta }_{jm}{\delta }_{k\ell }$ in suffix notation:Thus the cross product contraction expands as

Put $\tau =a\cdot (b\times c)$. The vector triple product identity givesCyclic invariance of the scalar triple product gives $(a\times b)\cdot c=\tau$. Hence the squared scalar triple product from cyclic cross products isIf an expansion entirely in pairwise dot products is desired, expand the Gram determinant of $a,b,c$:Both expressions remain valid for degenerate, including linearly dependent, vectors.

The parametric equation of a straight line through $a$ with unit direction $\hat{t}$ isIf the two lines meet, then $a_1+s{\hat{t}}_1=a_2+t{\hat{t}}_2$ for some real $s,t$, so $a_1-a_2$ lies in the linear span of their direction vectors. Its dot product with their cross product is consequently zero:This is not sufficient without excluding parallel lines. For instance $a_1=(0,0,0)$, $a_2=(0,1,0)$ and ${\hat{t}}_1={\hat{t}}_2=(1,0,0)$ satisfy it, but the lines are distinct and never meet. For nonparallel lines the condition is sufficient, because the two directions span the plane perpendicular to their nonzero cross product.

For nonparallel skew lines, put $n=({\hat{t}}_1\times {\hat{t}}_2)/|{\hat{t}}_1\times {\hat{t}}_2|$. Every joining vector is $a_1-a_2+s{\hat{t}}_1-t{\hat{t}}_2$, with fixed component $(a_1-a_2)\cdot n$ along $n$. Its length is at least the absolute value of that component. Its perpendicular component can be canceled by a unique choice of $s,t$, since the two independent directions span $n^{\perp }$. This lower bound is attained, proving the distance between skew lines: