# 2018 MJC J2 MYE Q6 (b) Modified

## Timothy Gan

##### 2018 MJC J2 MYE Q6 (b) Modified

A point $A$ with position vector $\overrightarrow{OA}=\alpha \,\mathbf{i}+\beta \,\mathbf{j}+\gamma \,\mathbf{k},$ where $\alpha ,\,\beta$ and $\gamma$ are real constants, has direction cosines $\cos \theta ,\,\,\cos \phi$ and $\cos \delta ,$ where $\theta ,\,\phi$ and $\delta$ are the angles $\overrightarrow{OA}$ makes with the positive $x,\,y$ and $z$-axes respectively.

(i)

Express the direction cosines $\cos \theta ,\,\,\cos \phi$ and $\cos \delta$ in terms of $\alpha ,\,\beta$ and $\gamma .$ Hence find the value of ${{\cos }^{2}}\theta +{{\cos }^{2}}\phi +{{\cos }^{2}}\delta .$

(ii)

The vector $\mathbf{d}$ makes angle of $45{}^\circ$ with the $x$-axis, $60{}^\circ$ with the $y$-axis and $\delta$ with the $z$-axis, where $0{}^\circ \le \delta \le 90{}^\circ .$ Find the value of $\delta .$ If the magnitude of $\mathbf{d}$ is $12,$ express $\mathbf{d}$ in cartesian form.

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## Timothy Gan

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