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      <text>9.5.4.2 Plane equation</text>
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      <text><![CDATA[<h4>The vector equation of a plane</h4><p>The answer to the above question, <span class="math inline">\(\underline{ r} \cdot \underline{ n}=\underline{ a} \cdot \underline{ n},\)</span> is the <strong>equation of a plane</strong>, written in vector form, passing through <span class="math inline">\(A\)</span> and perpendicular to <span class="math inline">\(\underline{n}\)</span>.</p>
<p>In this formula it does not matter whether or not <span class="math inline">\(\underline{n}\)</span> is a unit vector. If <span class="math inline">\(\underline{ \hat n}\)</span> is a unit vector then <span class="math inline">\(\underline{ a} \cdot \underline{\hat n}\)</span> represents the perpendicular distance from the origin to the plane which we usually denote by <span class="math inline">\(d\)</span>. Hence we can write <span class="math inline">\(\underline{ r} \cdot \underline{ \hat n}= d.\)</span> </p><p>This is the <strong>equation of a plane</strong>, written in vector form, with unit normal <span class="math inline">\(\underline{\hat n}\)</span> and which is a perpendicular distance <span class="math inline">\(d\)</span> from <span class="math inline">\(O\)</span>.</p>
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<h4>Key Point</h4>
<p>A plane with unit normal <span class="math inline">\(\underline{ \hat n}\)</span>, which is a perpendicular distance <span class="math inline">\(d\)</span> from <span class="math inline">\(O\)</span> is given by <span class="math display">\[\underline{ r} \cdot \underline{ \hat n}=d.\]</span></p>
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