<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/1783577066686-nwkpkpnl7cj.png" style="max-width: min(180px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>The origin \(O\) and the points \(A\), \(B\) and \(C\) lie in the same plane, where&nbsp;\(\overrightarrow{OA} = \mathbf{a}\), \(\overrightarrow{OB} = \mathbf{b}\) and \(\overrightarrow{OC} = \mathbf{c}\). It is given that&nbsp;\(\angle AOC = \angle BOC = \theta\), where&nbsp;\(\theta\) is an acute angle.</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;\(\mathbf{c} \cdot \mathbf{\hat{a}} = \mathbf{c} \cdot \mathbf{\hat{b}}\) where&nbsp;\(\mathbf{\hat{a}}\) and \(\mathbf{\hat{b}}\) are unit vectors in the directions of vectors \(\mathbf{a}\) and \(\mathbf{b}\) respectively.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">If&nbsp;\(\mathbf{c}\) can be written as&nbsp;\(m\mathbf{\hat{a}} + n\mathbf{\hat{b}}\), where&nbsp;\(m\) and \(n\) are constants, use the result from <strong>(a)</strong> to show that&nbsp;\(m = n\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Write down the equation of the line passing through the points \(A\) and \(B\).<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Given that&nbsp;\(|\mathbf{a}| = 3\), \(|\mathbf{b}| = 2\) and \(m = n\), show that the position vector of the point of intersection of the line passing through \(A\) and \(B\) and the line passing through \(O\) and \(C\) is \(t\left(\mathbf{\hat{a}} + \mathbf{\hat{b}}\right)\), where&nbsp;\(t\) is a constant to be determined.<span class="inline-marks">[3]</span></div></li></ol>

The origin \(O\) and the points \(A\), \(B\) and \(C\) lie in the same plane, where \(\overrightarrow{OA} = \mathbf{a}\), \(\overrightarrow{OB} = \mathbf{b}\) and \(\overrightarrow{OC} = \mathbf{c}\). It is given that \(\angle AOC = \angle BOC = \theta\), where \(\theta\) is an acute angle.

Show that \(\mathbf{c} \cdot \mathbf{\hat{a}} = \mathbf{c} \cdot \mathbf{\hat{b}}\) where \(\mathbf{\hat{a}}\) and \(\mathbf{\hat{b}}\) are unit vectors in the directions of vectors \(\mathbf{a}\) and \(\mathbf{b}\) respectively.

If \(\mathbf{c}\) can be written as \(m\mathbf{\hat{a}} + n\mathbf{\hat{b}}\), where \(m\) and \(n\) are constants, use the result from **(a)** to show that \(m = n\).

Write down the equation of the line passing through the points \(A\) and \(B\).

Given that \(|\mathbf{a}| = 3\), \(|\mathbf{b}| = 2\) and \(m = n\), show that the position vector of the point of intersection of the line passing through \(A\) and \(B\) and the line passing through \(O\) and \(C\) is \(t\left(\mathbf{\hat{a}} + \mathbf{\hat{b}}\right)\), where \(t\) is a constant to be determined.

Vectors

International Baccalaureate Higher Level Mathematics

2024 ACJC P2 Q4

2024 ACJC P2 Q4