Standard +0.8 This is a multi-step 3D vector geometry problem requiring: (1) verification that A lies on l, (2) using perpendicularity condition to confirm the setup, (3) finding distance PA, (4) using the 45° angle condition with dot product or geometric reasoning to find two positions of B on the line. It requires solid understanding of vector geometry, angle conditions, and careful algebraic manipulation across multiple steps, making it moderately challenging but within reach for well-prepared Further Maths students.
With respect to a fixed origin \(O\), the line \(l\) has equation
$$\mathbf{r} = \begin{pmatrix} 13 \\ 8 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 2 \\ -1 \end{pmatrix}, \text{ where } \lambda \text{ is a scalar parameter.}$$
The point \(A\) lies on \(l\) and has coordinates \((3, -2, 6)\).
The point \(P\) has position vector \((-\mathbf{i} + 2\mathbf{k})\) relative to \(O\).
Given that vector \(\overrightarrow{PA}\) is perpendicular to \(l\), and that point \(B\) is a point on \(l\) such that \(\angle BPA = 45°\), find the coordinates of the two possible positions of \(B\). [5]
With respect to a fixed origin $O$, the line $l$ has equation
$$\mathbf{r} = \begin{pmatrix} 13 \\ 8 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 2 \\ -1 \end{pmatrix}, \text{ where } \lambda \text{ is a scalar parameter.}$$
The point $A$ lies on $l$ and has coordinates $(3, -2, 6)$.
The point $P$ has position vector $(-\mathbf{i} + 2\mathbf{k})$ relative to $O$.
Given that vector $\overrightarrow{PA}$ is perpendicular to $l$, and that point $B$ is a point on $l$ such that $\angle BPA = 45°$, find the coordinates of the two possible positions of $B$. [5]
\hfill \mbox{\textit{SPS SPS FM Pure 2022 Q8 [5]}}