<p>Two planes \(p_1\) and \(p_2\) have respective cartesian equations given by</p>
<p style="text-align: center">\[4x + y + z = 8 \text{ and } 4x + 3y - z = 0.\]</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the sine of the acute angle between \(p_1\) and \(p_2\) in the form \(\frac{m}{\sqrt{n}}\) where \(m\) and \(n\) are positive integers to be determined.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Verify that the point \(A\) with coordinates \((1,0,4)\) lies on \(p_1\) and \(p_2\). Hence, without the use of a calculator, find the vector equation of the line \(l\) formed by the intersection of \(p_1\) and \(p_2\).</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">It is given that \(B\) is a point on \(p_1\) with coordinates \((1,3,1)\). Show that \(AB\) is perpendicular to \(l\) and hence use (a) to deduce exactly the shortest distance from \(B\) to \(p_2\).</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the cartesian equation of the plane \(p_3\) which contains \(B\) and is perpendicular to both \(p_1\) and \(p_2\).</span><span class="inline-marks">[2]</span></div></li></ol>

Two planes \(p_1\) and \(p_2\) have respective cartesian equations given by

\[4x + y + z = 8 \text{ and } 4x + 3y - z = 0.\]

Find the sine of the acute angle between \(p_1\) and \(p_2\) in the form \(\frac{m}{\sqrt{n}}\) where \(m\) and \(n\) are positive integers to be determined.

Verify that the point \(A\) with coordinates \((1,0,4)\) lies on \(p_1\) and \(p_2\). Hence, without the use of a calculator, find the vector equation of the line \(l\) formed by the intersection of \(p_1\) and \(p_2\).

It is given that \(B\) is a point on \(p_1\) with coordinates \((1,3,1)\). Show that \(AB\) is perpendicular to \(l\) and hence use (a) to deduce exactly the shortest distance from \(B\) to \(p_2\).

Find the cartesian equation of the plane \(p_3\) which contains \(B\) and is perpendicular to both \(p_1\) and \(p_2\).

2025 NYJC P2 Q5

2025 NYJC P2 Q5