Standard +0.3 This is a standard FP3 vector geometry question testing the relationship between a line and plane. It requires checking if the direction vector is perpendicular to the normal (for parallel/lying in plane) and substituting a point to verify. While it involves multiple steps, the techniques are routine and well-practiced, making it slightly easier than average for Further Maths content.
A line \(l\) has equation \(\mathbf{r} = 3\mathbf{i} + \mathbf{j} - 2\mathbf{k} + t(\mathbf{i} + 4\mathbf{j} + 2\mathbf{k})\) and a plane \(\Pi\) has equation \(8x - 7y + 10z = 7\). Determine whether \(l\) lies in \(\Pi\), is parallel to \(\Pi\) without intersecting it, or intersects \(\Pi\) at one point. [5]
A line $l$ has equation $\mathbf{r} = 3\mathbf{i} + \mathbf{j} - 2\mathbf{k} + t(\mathbf{i} + 4\mathbf{j} + 2\mathbf{k})$ and a plane $\Pi$ has equation $8x - 7y + 10z = 7$. Determine whether $l$ lies in $\Pi$, is parallel to $\Pi$ without intersecting it, or intersects $\Pi$ at one point. [5]
\hfill \mbox{\textit{OCR FP3 Q2 [5]}}