<p>The planes \(p\) and \(q\) have equations</p>
<p style="text-align: center">\[\mathbf{r} = (2 + \lambda - 2\mu)\mathbf{i} + (-3\lambda + \mu)\mathbf{j} + (3 - \lambda + 2\mu)\mathbf{k} \quad \text{and} \quad \mathbf{r} \cdot \begin{pmatrix} a \\ -1 \\ b \end{pmatrix} = 1 \text{ respectively,}\]</p>
<p>where&nbsp;\(a\) and \(b\) are constants and \(\lambda\) and \(\mu\) are parameters.</p>
<p>The line \(l\) passes through the point \((5, 4, 0)\) and is parallel to the vector \(-2\mathbf{i} - \mathbf{j} + 2\mathbf{k}\). The planes \(p\) and \(q\) meet in the line \(l\).</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that&nbsp;\(a = 1\) and \(b = \frac{1}{2}\).<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find the exact acute angle between the planes \(p\) and \(q\).<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Find the distance from the point \(A(2, 0, 3)\) to the plane \(q\). Hence deduce the shortest distance from \(A\) to \(l\).<span class="inline-marks">[4]</span></div></li></ol>
<p>The plane \(q\) is reflected about the plane \(p\) to obtain the plane \(q'\).</p>
<ol class="abc-list" start="4"><li><div class="list-item-paragraph">Find a cartesian equation of the plane \(q'\).<span class="inline-marks">[3]</span></div></li></ol>

The planes \(p\) and \(q\) have equations

\[\mathbf{r} = (2 + \lambda - 2\mu)\mathbf{i} + (-3\lambda + \mu)\mathbf{j} + (3 - \lambda + 2\mu)\mathbf{k} \quad \text{and} \quad \mathbf{r} \cdot \begin{pmatrix} a \\ -1 \\ b \end{pmatrix} = 1 \text{ respectively,}\]

where \(a\) and \(b\) are constants and \(\lambda\) and \(\mu\) are parameters.

The line \(l\) passes through the point \((5, 4, 0)\) and is parallel to the vector \(-2\mathbf{i} - \mathbf{j} + 2\mathbf{k}\). The planes \(p\) and \(q\) meet in the line \(l\).

Show that \(a = 1\) and \(b = \frac{1}{2}\).

Find the exact acute angle between the planes \(p\) and \(q\).

Find the distance from the point \(A(2, 0, 3)\) to the plane \(q\). Hence deduce the shortest distance from \(A\) to \(l\).

2024 RI Promo Q9

2024 RI Promo Q9