<p>Referred to the origin \(O\), the points \(A\), \(B\) and \(C\) have position vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) respectively and they lie on plane \(\pi\) .</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Show that&nbsp;\(\mathbf{a}\times\mathbf{b} + \mathbf{b}\times\mathbf{c} + \mathbf{c}\times\mathbf{a}\) is a vector perpendicular to the plane \(\pi\) .<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Prove that the equation of plane \(\pi\) can be written as</div><div class="list-item-paragraph" style="text-align: center">\[\mathbf{r}\bullet(\mathbf{a}\times\mathbf{b} + \mathbf{b}\times\mathbf{c} + \mathbf{c}\times\mathbf{a}) = \mathbf{a}\bullet(\mathbf{b}\times\mathbf{c}),\]</div><div class="list-item-paragraph">explaining clearly the reason for any result that you use in your proof.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Given that&nbsp;\(\mathbf{a} = \mathbf{i}\), \(\mathbf{b} = \mathbf{j}\) and \(\mathbf{c} = \mathbf{k}\), show that the equation of the plane \(\pi\) can be written as&nbsp;\(\mathbf{r}\bullet\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} = 1\). Hence, find the cartesian equations of the planes which are at a distance of 5 units from plane \(\pi\).<span class="inline-marks">[3]</span></div></li></ol>

Referred to the origin \(O\), the points \(A\), \(B\) and \(C\) have position vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) respectively and they lie on plane \(\pi\) .

Show that \(\mathbf{a}\times\mathbf{b} + \mathbf{b}\times\mathbf{c} + \mathbf{c}\times\mathbf{a}\) is a vector perpendicular to the plane \(\pi\) .

Prove that the equation of plane \(\pi\) can be written as

\[\mathbf{r}\bullet(\mathbf{a}\times\mathbf{b} + \mathbf{b}\times\mathbf{c} + \mathbf{c}\times\mathbf{a}) = \mathbf{a}\bullet(\mathbf{b}\times\mathbf{c}),\]

explaining clearly the reason for any result that you use in your proof.

Given that \(\mathbf{a} = \mathbf{i}\), \(\mathbf{b} = \mathbf{j}\) and \(\mathbf{c} = \mathbf{k}\), show that the equation of the plane \(\pi\) can be written as \(\mathbf{r}\bullet\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} = 1\). Hence, find the cartesian equations of the planes which are at a distance of 5 units from plane \(\pi\).

Vectors

International Baccalaureate Higher Level Mathematics

2024 ACJC P2 Q2

2024 ACJC P2 Q2