With reference to the origin $O$, the points $A$ and $B$ have coordinates \(\left( 1,2,0 \righ

Question

With reference to the origin $O$, the points $A$ and $B$ have coordinates $\left( 1,2,0 \right)$ and $\left( -5,1,0 \right)$. Find the angle $AOB$.[3] The vectors $\overrightarrow{OA}$ and $\overrightarrow{OB}$ are projected along the direction of $\left( \begin{matrix} 1 \ 2 \ 2 \ \end{matrix} \right)$ and $\left( \begin{matrix} 2 \ 1 \ 2 \ \end{matrix} \right)$ to form the vectors $\overrightarrow{O'A'}$ and $\overrightarrow{O'B'}$.[3] Find the vectors $\overrightarrow{O'A'}$ and $\overrightarrow{O'B'}$.[3] Find the exact area of the triangle $O'A'B'$.[2]

Accepted Answer

(a) $105.3^\circ$ (b) $\overrightarrow{O'A'}=\dfrac59\begin{pmatrix}1\2\2\end{pmatrix}$, $\overrightarrow{O'B'}=-\begin{pmatrix}2\1\2\end{pmatrix}$ (c) $\dfrac{5\sqrt{17}}{18}\text{ units}^2$

2023 DHS Promo Q5

2023 DHS Promo Q5

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