2021 ACJC Promo Q6

The Cartesian equation of line \({{L}_{1}}\) is \(\frac{x-2}{a}=\frac{y+2}{b}=\frac{z-3}{c}\), where \(a\), \(b\), \(c\) are constants.

The line \({{L}_{2}}\) is parallel to the vector \(4\mathbf{i}+3\mathbf{j}\). The line \({{L}_{3}}\) passes through the origin and the point with position vector \(\mathbf{j}+\mathbf{k}\).

1. Given that \({{L}_{1}}\) is perpendicular to \({{L}_{2}}\), form an equation relating \(a\) and \(b\).[1]
2. Given that \({{L}_{1}}\) intersects \({{L}_{3}}\), show that \(5a+2b-2c=0\).[3]
3. Hence express \(a\) and \(b\) in terms of \(c\).[1]
4. Find the acute angle between \({{L}_{1}}\) and \({{L}_{3}}\).[2]