<p>\(\mathbf{v}\) and \(\mathbf{n}\) are unit vectors in \(3\)-dimensional space, with \(\mathbf{v} \cdot \mathbf{n} \neq 0\).</p>
<p>Let&nbsp;\(P\) be a plane with unit normal \(\mathbf{n}\). The reflection of \(\mathbf{v}\) in plane \(P\), \(\mathbf{w}\), is given by the formula\n\[\mathbf{w} = \mathbf{v} - 2(\mathbf{v} \cdot \mathbf{n})\mathbf{n} . \quad (*)\]</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Calculate the vector \(\mathbf{w}\) when&nbsp;\(\mathbf{v} = \begin{pmatrix} 0 \\ 0 \\ -1 \end{pmatrix}\) and \(\mathbf{n} = \frac{1}{5} \begin{pmatrix} 3 \\ 0 \\ 4 \end{pmatrix}\).<span class="inline-marks">[2]</span></div></li></ol>
<p>When the unit vectors \(\mathbf{v}\) and \(\mathbf{n}\) vary, the formula \((*)\) will always result in a unit vector for&nbsp;\(\mathbf{w}\).</p>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">Verify that the vector \(\mathbf{w}\) calculated in part <strong>(a)</strong> is a unit vector.<span class="inline-marks">[1]</span></div></li></ol>
<p>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 \(\mathbf{n}\).</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1779451489366-7psqn7e00y7.png" style="max-width: min(200px, 100%); height: auto; display: block;" alt="Solar plant diagram showing incoming beam of sunlight reflecting off a mirror at point M towards a receiver at point C." class="aligncenter custom-width" /></figure></div>
<ol class="abc-list" start="3"><li><div class="list-item-paragraph">Write down the unit vector \(\mathbf{w}\) in the direction of \(\overline{MC}\).<span class="inline-marks">[1]</span></div></li></ol>
<p>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 \(C\).</p>
<ol class="abc-list" start="4"><li><div class="list-item-paragraph">Using the formula \((*)\) or otherwise, derive that a unit normal for plane \(P\) is \(\mathbf{n} = \frac{1}{\sqrt{30}} \begin{pmatrix} 2 \\ 1 \\ 5 \end{pmatrix}\).<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Hence find a cartesian equation of plane \(P\).<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find the acute angle that plane \(P\) makes with the \(xy\)-plane.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">State, in the context of the question, an interpretation of the condition \(\mathbf{v} \cdot \mathbf{n} \neq 0\).<span class="inline-marks">[1]</span></div></li></ol>

\(\mathbf{v}\) 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\n\[\mathbf{w} = \mathbf{v} - 2(\mathbf{v} \cdot \mathbf{n})\mathbf{n} . \quad (*)\]

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

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

Verify that the vector \(\mathbf{w}\) calculated in part **(a)** is a unit vector.

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 \(\mathbf{n}\).

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

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 \(C\).

Using the formula \((*)\) or otherwise, derive that a unit normal for plane \(P\) is \(\mathbf{n} = \frac{1}{\sqrt{30}} \begin{pmatrix} 2 \\ 1 \\ 5 \end{pmatrix}\).

Hence find a cartesian equation of plane \(P\).

Find the acute angle that plane \(P\) makes with the \(xy\)-plane.

State, in the context of the question, an interpretation of the condition \(\mathbf{v} \cdot \mathbf{n} \neq 0\).

2024 EJC Promo Q13

2024 EJC Promo Q13