Paper 1, Section II, 5B

Vectors and Matrices | Part IA, 2014

(i) For vectors a,b,cR3\mathbf{a}, \mathbf{b}, \mathbf{c} \in \mathbb{R}^{3}, show that

a×(b×c)=(ac)b(ab)c.\mathbf{a} \times(\mathbf{b} \times \mathbf{c})=(\mathbf{a} \cdot \mathbf{c}) \mathbf{b}-(\mathbf{a} \cdot \mathbf{b}) \mathbf{c} .

Show that the plane (ra)n=0(\mathbf{r}-\mathbf{a}) \cdot \mathbf{n}=0 and the line (rb)×m=0(\mathbf{r}-\mathbf{b}) \times \mathbf{m}=\mathbf{0}, where mn0\mathbf{m} \cdot \mathbf{n} \neq 0, intersect at the point

r=(an)m+n×(b×m)mn\mathbf{r}=\frac{(\mathbf{a} \cdot \mathbf{n}) \mathbf{m}+\mathbf{n} \times(\mathbf{b} \times \mathbf{m})}{\mathbf{m} \cdot \mathbf{n}}

and only at that point. What happens if mn=0\mathbf{m} \cdot \mathbf{n}=0 ?

(ii) Explain why the distance between the planes (ra1)n^=0\left(\mathbf{r}-\mathbf{a}_{1}\right) \cdot \hat{\mathbf{n}}=0 and (ra2)n^=0\left(\mathbf{r}-\mathbf{a}_{2}\right) \cdot \hat{\mathbf{n}}=0 is (a1a2)n^\left|\left(\mathbf{a}_{1}-\mathbf{a}_{2}\right) \cdot \hat{\mathbf{n}}\right|, where n^\hat{\mathbf{n}} is a unit vector.

(iii) Find the shortest distance between the lines (3+s,3s,4s)(3+s, 3 s, 4-s) and (2,3+t,3t)(-2,3+t, 3-t) where s,tRs, t \in \mathbb{R}. [You may wish to consider two appropriately chosen planes and use the result of part (ii).]

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