Let \(t\) be a positive constant.

The line \(l_1\) passes through the point with position vector \(t\mathbf{i} + \mathbf{j}\) and is parallel to the vector \(-2\mathbf{i} - \mathbf{j}\).

The line \(l_2\) passes through the point with position vector \(\mathbf{j} + t\mathbf{k}\) and is parallel to the vector \(-2\mathbf{j} + \mathbf{k}\).

It is given that the shortest distance between the lines \(l_1\) and \(l_2\) is \(\sqrt{21}\).

(a) Find the value of \(t\). [5]

The plane \(\Pi_1\) contains \(l_1\) and is parallel to \(l_2\).

(b) Write down an equation of \(\Pi_1\), giving your answer in the form \(\mathbf{r} = \mathbf{a} + \lambda \mathbf{b} + \mu \mathbf{c}\).

The plane \(\Pi_2\) has Cartesian equation \(5x - 6y + 7z = 0\).

(c) Find the acute angle between \(l_2\) and \(\Pi_2\). [3]

(d) Find the acute angle between \(\Pi_1\) and \(\Pi_2\). [3]