7. Line \(l _ { 1 }\) has Cartesian equation
$$x - 3 = \frac { 2 y + 2 } { 3 } = 2 - z$$
- Write the equation of line \(l _ { 1 }\) in the form
$$\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }$$
where \(\lambda\) is a parameter and \(\mathbf { a }\) and \(\mathbf { b }\) are vectors to be found.
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[2 marks] - Line \(l _ { 2 }\) passes through the points \(P ( 3,2,0 )\) and \(Q ( n , 5 , n )\), where \(n\) is a constant.
- Show that the lines \(l _ { 1 }\) and \(l _ { 2 }\) are not perpendicular.
- (ii) Explain briefly why lines \(l _ { 1 }\) and \(l _ { 2 }\) cannot be parallel.
- (iii) Given that \(\theta\) is the acute angle between lines \(l _ { 1 }\) and \(l _ { 2 }\), show that
$$\cos \theta = \frac { p } { \sqrt { 34 n ^ { 2 } + q n + 306 } }$$
where \(p\) and \(q\) are constants to be found.
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[ADDITIONAL SPACE FOR QUESTION 7]
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