Paper 1, Section II, 5C5 \mathrm{C}

Vectors and Matrices | Part IA, 2011

Explain why each of the equations

x=a+λbx×c=d\begin{aligned} &\mathbf{x}=\mathbf{a}+\lambda \mathbf{b} \\ &\mathbf{x} \times \mathbf{c}=\mathbf{d} \end{aligned}

describes a straight line, where a,b,c\mathbf{a}, \mathbf{b}, \mathbf{c} and d\mathbf{d} are constant vectors in R3,b\mathbb{R}^{3}, \mathbf{b} and c\mathbf{c} are non-zero, cd=0\mathbf{c} \cdot \mathbf{d}=0 and λ\lambda is a real parameter. Describe the geometrical relationship of a, b,c\mathbf{b}, \mathbf{c} and d\mathbf{d} to the relevant line, assuming that d0\mathbf{d} \neq \mathbf{0}.

Show that the solutions of (2) satisfy an equation of the form (1), defining a,b\mathbf{a}, \mathbf{b} and λ(x)\lambda(\mathbf{x}) in terms of c\mathbf{c} and d\mathbf{d} such that ab=0\mathbf{a} \cdot \mathbf{b}=0 and b=c|\mathbf{b}|=|\mathbf{c}|. Deduce that the conditions on c\mathbf{c} and d\mathbf{d} are sufficient for (2) to have solutions.

For each of the lines described by (1) and (2), find the point x\mathbf{x} that is closest to a given fixed point y\mathbf{y}.

Find the line of intersection of the two planes xm=μ\mathbf{x} \cdot \mathbf{m}=\mu and xn=ν\mathbf{x} \cdot \mathbf{n}=\nu, where m\mathbf{m} and n\mathbf{n} are constant unit vectors, m×n0\mathbf{m} \times \mathbf{n} \neq \mathbf{0} and μ\mu and ν\nu are constants. Express your answer in each of the forms (1) and (2), giving both a\mathbf{a} and d\mathbf{d} as linear combinations of m\mathbf{m} and n\mathbf{n}.

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