Geometric proofs using vectors
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The position vectors of two points \(A\) and \(B\) relative to an origin \(O\) are \(2i+4j\) and \(4i-2j\) respectively.
Given that \(\overrightarrow {OC} = \dfrac{1}{2}\overrightarrow {OB} \) and \(\overrightarrow {AD} = \dfrac{1}{4}\overrightarrow {AB} \), find
i) \(\overrightarrow {OC} \) and \(\overrightarrow {OD} \)
ii) Hence find \(\left| {\overrightarrow {CD} } \right|\)
i) \(\overrightarrow {OC}=2i-j \) and \(\overrightarrow {OD}=\dfrac{5}{2}i+\dfrac{5}{2}j \)
ii) \(\left| {\overrightarrow {CD} } \right|=\dfrac{5 \sqrt{2}}{2}\)
$$\begin{align}
&\begin{aligned}\text{i)}\quad
\overrightarrow{OA}&=2i+4 j,\\
\overrightarrow{OB}&=4 i-2 j\\
\overrightarrow{OC}&=\frac{1}{2} \overrightarrow{OB}\\
&=\frac{1}{2}(4 i-2 j)\\
&=2 i-j\\
\overrightarrow{A B}&=\overrightarrow{A O}+\overrightarrow{O B}\\
&=-(2 i+4 j)+4 i-2 j\\
&=2i-6j\\
\overrightarrow{A D}&=\frac{1}{4} \overrightarrow{A B}\\
&=\frac{1}{4}(2 i-6 i)\\
&=\frac{1}{2} i-\frac{3}{2}j\\
\overrightarrow{O D}&=\overrightarrow{OA}+\overrightarrow{A D}\\
&=2i+4j+\frac{1}{2}i-\frac{3}{2}j\\
&=\frac{5}{2}i+\frac{5}{2}j\\
\therefore\ \overrightarrow{O C}&=2i-j,\quad \overrightarrow{OD}=\frac{5}{2}i+\frac{5}{2}j\end{aligned}\\
&\begin{aligned}\text{ii)}\quad
\overrightarrow{C D} &=\overrightarrow{C O}+\overrightarrow{O D} \\
&=-2 i+j+\frac{5}{2} i+\frac{5}{2} j \\
&=\frac{1}{2} i+\frac{7}{2} j \\
|\overrightarrow{C D}| &=\sqrt{\left(\frac{1}{2}\right)^{2}+\left(\frac{7}{2}\right)^{2}} \\
&=\sqrt{\frac{1}{4}+\frac{49}{4}} \\
&=\frac{\sqrt{50}}{2} \\
&=\frac{5 \sqrt{2}}{2}
\end{aligned}
\end{align}$$
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