A point $A$ with position vector $\overrightarrow{OA}=\alpha \,\mathbf{i}+\beta \,\mathbf{j}+\gamma

Question

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 	heta ,\,\,\cos \phi $ and $\cos \delta ,$ where $	heta ,\,\phi $ and $\delta $ are the angles $\overrightarrow{OA}$ makes with the positive $x,\,y$ and $z$-axes respectively. Express the direction cosines $\cos 	heta ,\,\,\cos \phi $ and $\cos \delta $ in terms of $\alpha ,\,\beta $ and $\gamma .$ Hence find the value of ${{\cos }^{2}}	heta +{{\cos }^{2}}\phi +{{\cos }^{2}}\delta .$ 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.

Accepted Answer

(i) cos²θ + cos²φ + cos²δ = 1; (ii) d = 6√2i + 6j + 6k

2018 MJC J2 MYE Q6 (b) Modified

2018 MJC J2 MYE Q6 (b) Modified

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