2024 EJC Promo Q13

\(\mathbf{v}\) and \(\mathbf{n}\) are unit vectors in \(3\)-dimensional space, with \(\mathbf{v} \cdot \mathbf{n} \neq 0\).

Let \(P\) be a plane with unit normal \(\mathbf{n}\). The reflection of \(\mathbf{v}\) in plane \(P\), \(\mathbf{w}\), is given by the formula\[\mathbf{w} = \mathbf{v} - 2(\mathbf{v} \cdot \mathbf{n})\mathbf{n} . \quad (*)\]

1. Calculate the vector \(\mathbf{w}\) when \(\mathbf{v} = \begin{pmatrix} 0 \\ 0 \\ -1 \end{pmatrix}\) and \(\mathbf{n} = \frac{1}{5} \begin{pmatrix} 3 \\ 0 \\ 4 \end{pmatrix}\).[2]

When the unit vectors \(\mathbf{v}\) and \(\mathbf{n}\) vary, the formula \((*)\) will always result in a unit vector for \(\mathbf{w}\).

2. Verify that the vector \(\mathbf{w}\) calculated in part (a) is a unit vector.[1]

Before photovoltaic technology allowed for cost-effective direct conversion of sunlight to electrical energy in solar panels, solar power plants relied on arrays of mirrors to concentrate a large area of sunlight into a receiver. In one such plant (see diagram), the receiver is modelled as a single point \(C\) with coordinates \((0, 0, 10)\), with mirrors surrounding it. One particular mirror has centre \(M\) with coordinates \((-10, -5, 0)\), and its surface is modelled as part of a plane \(P\) passing through \(M\) with unit normal \(\mathbf{n}\).

Solar plant diagram showing incoming beam of sunlight reflecting off a mirror at point M towards a receiver at point C.

3. Write down the unit vector \(\mathbf{w}\) in the direction of \(\overline{MC}\).[1]

At midday, incoming beams of sunlight can be modelled as travelling along the direction \(\mathbf{v} = \begin{pmatrix} 0 \\ 0 \\ -1 \end{pmatrix}\). The mirror is angled so that a beam of sunlight shining on \(M\) is reflected to pass through the receiver \(C\).

4. Using the formula \((*)\) or otherwise, derive that a unit normal for plane \(P\) is \(\mathbf{n} = \frac{1}{\sqrt{30}} \begin{pmatrix} 2 \\ 1 \\ 5 \end{pmatrix}\).[3]
5. Hence find a cartesian equation of plane \(P\).[2]
6. Find the acute angle that plane \(P\) makes with the \(xy\)-plane.[2]
7. State, in the context of the question, an interpretation of the condition \(\mathbf{v} \cdot \mathbf{n} \neq 0\).[1]