- [Maximum mark: 8]
A line (L_1) has vector equation (\mathbf{r} = \begin{pmatrix} 0 \ 0 \ 2 \end{pmatrix} + t \begin{pmatrix} 1 \ 0 \ 1 \end{pmatrix}) where (t \in \mathbb{R}).
The plane (\Pi_1) contains the line (L_1) and passes through the point (2, 1, 5).
(a) Show that the Cartesian equation of the plane (\Pi_1) is (x + y - z = -2). [4]
Consider the three planes:
(\Pi_1: x + y - z = -2)
(\Pi_2: 2x + by - z = 3)
(\Pi_3: x - y + 2z = d)
where (b, d \in \mathbb{Q}^+).
The three planes intersect in a line.
(b) Find the value of (b) and the value of (d). [4]