FP3 June 2014 Q1

1. The line \(l\) passes through the point \(P(2, 1, 3)\) and is perpendicular to the plane \(\Pi\) whose vector equation is \[\mathbf{r} \cdot (\mathbf{i} - 2\mathbf{j} - \mathbf{k}) = 3\]

Find

(a) a vector equation of the line \(l\), (2)

(b) the position vector of the point where \(l\) meets \(\Pi\). (4)

Mark scheme (a)

Scheme	Marks
\(P(2, 1, 3)\) and \(\mathbf{r} \cdot (\mathbf{i} - 2\mathbf{j} - \mathbf{k}) = 3\)
\(l\) is parallel to \(\mathbf{i} - 2\mathbf{j} - \mathbf{k}\) An appreciation that \(\mathbf{i} - 2\mathbf{j} - \mathbf{k}\) is the direction of the line (may be implied).	M1
"\(\mathbf{r}\)" \(= 2\mathbf{i} + \mathbf{j} + 3\mathbf{k} + t(\mathbf{i} - 2\mathbf{j} - \mathbf{k})\) or \((\mathbf{r} - (2\mathbf{i} + \mathbf{j} + 3\mathbf{k})) \times (\mathbf{i} - 2\mathbf{j} - \mathbf{k}) = \mathbf{0}\) or \(\mathbf{r} \times (\mathbf{i} - 2\mathbf{j} - \mathbf{k}) = (2\mathbf{i} + \mathbf{j} + 3\mathbf{k}) \times (\mathbf{i} - 2\mathbf{j} - \mathbf{k})\) or \(\mathbf{r} \times (\mathbf{i} - 2\mathbf{j} - \mathbf{k}) = 5\mathbf{i} + 5\mathbf{j} - 5\mathbf{k}\) A correct vector equation* in any form. (Allow any multiple of the direction vector.)*	A1
	(2)

Mark scheme (b)

Scheme	Marks
\(\begin{pmatrix}2 + t \\ 1 - 2t \\ 3 - t\end{pmatrix}\begin{pmatrix}1 \\ -2 \\ -1\end{pmatrix} = 3\) Substitutes a parametric form of their line from part (a) into the equation of the plane. This statement is sufficient.	M1
\(2 + t - 2(1 - 2t) - (3 - t) = 3\) Correct equation (allow unsimplified)	A1
\(2 + t - 2 + 4t - 3 + t = 3 \Rightarrow t = \ldots\ldots\) Solves to find a value for \(t\) Dependent on the first M	dM1
\(t = 1 \Rightarrow l\) meets \(\Pi\) at \(\mathbf{r} = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k}\) Correct position vector (allow as coordinates (3, -1, 2))	A1
	(4)

Mark scheme (c)

Scheme	Marks
\(PQ = \|3\mathbf{i} - \mathbf{j} + 2\mathbf{k} - (2\mathbf{i} + \mathbf{j} + 3\mathbf{k})\|\)
\(= \|\mathbf{i} - 2\mathbf{j} - \mathbf{k}\| = \sqrt{1^2 + 2^2 + 1^2}\) Attempts the vector \(PQ\) or \(QP\) and correct Pythagoras.	M1
\(= \sqrt{6}\) Allow awrt 2.45 or \(\dfrac{6}{\sqrt{6}}\)	A1
	(2)
	(8 marks)

Notes

(c) Way 2

Scheme	Marks
\(t = \text{"}1\text{"} \Rightarrow \overrightarrow{PQ} = \text{"}1\text{"} \times (\mathbf{i} - 2\mathbf{j} - \mathbf{k})\)
\(= \|\mathbf{i} - 2\mathbf{j} - \mathbf{k}\| = \sqrt{1^2 + 2^2 + 1^2}\) Attempts the vector \(PQ\) using their value for \(t\) and their normal vector and correct Pythagoras.	M1
\(= \sqrt{6}\) Allow awrt 2.45 or \(\dfrac{6}{\sqrt{6}}\)	A1
	(2)