Paper 1, Section II, C

Vectors and Matrices | Part IA, 2020

(a) Let A,BA, B, and CC be three distinct points in the plane R2\mathbb{R}^{2} which are not collinear, and let a,b\mathbf{a}, \mathbf{b}, and c\mathbf{c} be their position vectors.

Show that the set LABL_{A B} of points in R2\mathbb{R}^{2} equidistant from AA and BB is given by an equation of the form

nABx=pAB,\mathbf{n}_{A B} \cdot \mathbf{x}=p_{A B},

where nAB\mathbf{n}_{A B} is a unit vector and pABp_{A B} is a scalar, to be determined. Show that LABL_{A B} is perpendicular to AB\overrightarrow{A B}.

Show that if x\mathbf{x} satisfies

nABx=pAB and nBCx=pBC\mathbf{n}_{A B} \cdot \mathbf{x}=p_{A B} \quad \text { and } \quad \mathbf{n}_{B C} \cdot \mathbf{x}=p_{B C}


nCAx=pCA.\mathbf{n}_{C A} \cdot \mathbf{x}=p_{C A} .

How do you interpret this result geometrically?

(b) Let a\mathbf{a} and u\mathbf{u} be constant vectors in R3\mathbb{R}^{3}. Explain why the vectors x\mathbf{x} satisfying

x×u=a×u\mathbf{x} \times \mathbf{u}=\mathbf{a} \times \mathbf{u}

describe a line in R3\mathbb{R}^{3}. Find an expression for the shortest distance between two lines x×uk=ak×uk\mathbf{x} \times \mathbf{u}_{k}=\mathbf{a}_{k} \times \mathbf{u}_{k}, where k=1,2k=1,2.

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