Paper 1, Section II, A

Vectors and Matrices | Part IA, 2016

(a) Use suffix notation to prove that

a(b×c)=c(a×b)\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})=\mathbf{c} \cdot(\mathbf{a} \times \mathbf{b})

(b) Show that the equation of the plane through three non-colinear points with position vectors a,b\mathbf{a}, \mathbf{b} and c\mathbf{c} is

r(a×b+b×c+c×a)=a(b×c)\mathbf{r} \cdot(\mathbf{a} \times \mathbf{b}+\mathbf{b} \times \mathbf{c}+\mathbf{c} \times \mathbf{a})=\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})

where r\mathbf{r} is the position vector of a point in this plane.

Find a unit vector normal to the plane in the case a=(2,0,1),b=(0,4,0)\mathbf{a}=(2,0,1), \mathbf{b}=(0,4,0) and c=(1,1,2)\mathbf{c}=(1,-1,2).

(c) Let r\mathbf{r} be the position vector of a point in a given plane. The plane is a distance dd from the origin and has unit normal vector n\mathbf{n}, where nr0\mathbf{n} \cdot \mathbf{r} \geqslant 0. Write down the equation of this plane.

This plane intersects the sphere with centre at p\mathbf{p} and radius qq in a circle with centre at m\mathbf{m} and radius ρ\rho. Show that

mp=γn\mathbf{m}-\mathbf{p}=\gamma \mathbf{n}

Find γ\gamma in terms of qq and ρ\rho. Hence find ρ\rho in terms of n,d,p\mathbf{n}, d, \mathbf{p} and qq.

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