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.

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