RI Tutorial Vectors Planes Q2

A line \({l}\) is given by the equation \({\mathbf{r} = 2\mathbf{i} + \mathbf{k} + \lambda\left(\mathbf{i} + 2\mathbf{j} - 3\mathbf{k}\right)}\), \({\lambda \in \mathbb{R}}\) and the point \({P}\) has position vector \({2\mathbf{i} - \mathbf{j} - 3\mathbf{k}}\) relative to the origin. \({N}\) is a point on \({l}\) such that the line \({\overrightarrow{PN}}\) is perpendicular to \({l}\).

1. Show that \({\overrightarrow{PN} = \frac{1}{7}(5\mathbf{i} + 17\mathbf{j} + 13\mathbf{k})}\).[2]
2. The plane \({\Pi_1}\) has equation \({\mathbf{r} \cdot (3\mathbf{i} + \mathbf{k}) = 7}\). Verify that \({l}\) lies in \({\Pi_1}\), and find the perpendicular distance from point \({P}\) to \({\Pi_1}\). [4]
3. A second plane \({\Pi_2}\) contains \({l}\) and \({P}\). Using your answers above, or otherwise, determine the acute angle between \({\Pi_1}\) and \({\Pi_2}\). [2]
4. A third plane \({\Pi_3}\) has equation \({\mathbf{r} \cdot (-\mathbf{j} + 5\mathbf{k}) = 22}\). Determine the position vector of the point of intersection between planes \({\Pi_1}\), \({\Pi_2}\) and \({\Pi_3}\). [2]