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

Vectors and Matrices | Part IA, 2012

The equation of a plane Π\Pi in R3\mathbb{R}^{3} is

xn=d\mathbf{x} \cdot \mathbf{n}=d

where dd is a constant scalar and n\mathbf{n} is a unit vector normal to Π\Pi. What is the distance of the plane from the origin OO ?

A sphere SS with centre p\mathbf{p} and radius rr satisfies the equation


Show that the intersection of Π\Pi and SS contains exactly one point if pnd=r|\mathbf{p} \cdot \mathbf{n}-d|=r.

The tetrahedron OABCO A B C is defined by the vectors a=OA,b=OB\mathbf{a}=\overrightarrow{O A}, \mathbf{b}=\overrightarrow{O B}, and c=OC\mathbf{c}=\overrightarrow{O C}with a(b×c)>0\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})>0. What does the condition a(b×c)>0\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})>0 imply about the set of vectors {a,b,c}\{\mathbf{a}, \mathbf{b}, \mathbf{c}\} ? A sphere TrT_{r} with radius r>0r>0 lies inside the tetrahedron and intersects each of the three faces OAB,OBCO A B, O B C, and OCAO C A in exactly one point. Show that the centre PP of TrT_{r} satisfies

OP=rb×ca+c×ab+a×bca(b×c)\overrightarrow{O P}=r \frac{|\mathbf{b} \times \mathbf{c}| \mathbf{a}+|\mathbf{c} \times \mathbf{a}| \mathbf{b}+|\mathbf{a} \times \mathbf{b}| \mathbf{c}}{\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})}

Given that the vector a×b+b×c+c×a\mathbf{a} \times \mathbf{b}+\mathbf{b} \times \mathbf{c}+\mathbf{c} \times \mathbf{a} is orthogonal to the plane Ψ\Psi of the face ABCA B C, obtain an equation for Ψ\Psi. What is the distance of Ψ\Psi from the origin?

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