June 2025 Paper 2 Q9

OCR MEICurrent spec3 marksVectors

9 The position vectors of the points \(A\) and \(B\) are \(\overrightarrow{OA} = \begin{pmatrix}-1\\-2\\0\end{pmatrix}\) and \(\overrightarrow{OB} = \begin{pmatrix}2\\1\\-3\end{pmatrix}\) respectively.

(a) Write down the vector \(\overrightarrow{AB}\). [1]

The point \(C\) lies on \(AB\) such that \(2\overrightarrow{AC} = \overrightarrow{CB}\).

(b) Find the vector \(\overrightarrow{OC}\). [2]

Scheme	Marks	AO
\(\begin{pmatrix}3\\3\\-3\end{pmatrix}\) or \(3\mathbf{i} + 3\mathbf{j} - 3\mathbf{k}\)	B1	1.1
	[1]

Mark scheme (b)

Scheme	Marks	AO
\([\overrightarrow{OC} =]\ \begin{pmatrix}-1\\-2\\0\end{pmatrix} + \frac{1}{3}\begin{pmatrix}3\\3\\-3\end{pmatrix}\) oe	M1	3.1a
\(\begin{pmatrix}0\\-1\\-1\end{pmatrix}\) or \(-\mathbf{j} - \mathbf{k}\)	A1	1.1
	[2]

Notes

M1: FT their \(\begin{pmatrix}3\\3\\-3\end{pmatrix}\)

A1: allow B2 for correct answer unsupported; mark the final answer
if M0 allow SC1 for \(\left[\begin{pmatrix}-1\\-2\\0\end{pmatrix} + \frac{2}{3}\begin{pmatrix}3\\3\\-3\end{pmatrix} =\right]\begin{pmatrix}1\\0\\-2\end{pmatrix}\)
do not allow \((0, -1, -1)\);
only penalise coordinate form once

Alternative method 1

Scheme	Marks
\([\overrightarrow{AC} =]\ \begin{pmatrix}a - -1\\b - -2\\c - 0\end{pmatrix}\ [\overrightarrow{CB} =]\ \begin{pmatrix}2-a\\1-b\\-3-c\end{pmatrix}\) \(2\overrightarrow{AC} = \overrightarrow{CB}\) equating coefficients: \(2a + 2 = 2 - a\) \(2b + 4 = 1 - b\) \(2c = -3 - c\)	M1
\(\begin{pmatrix}0\\-1\\-1\end{pmatrix}\) or \(-\mathbf{j} - \mathbf{k}\)	A1

may work with i, j, k notation

M1: allow one sign error only

A1: do not allow \((0, -1, -1)\);
only penalise coordinate form once

Alternative method 2

Scheme	Marks
\(2\overrightarrow{OC} - 2\begin{pmatrix}-1\\-2\\0\end{pmatrix} = \begin{pmatrix}2\\1\\-3\end{pmatrix} - \overrightarrow{OC}\)	M1
\(\begin{pmatrix}0\\-1\\-1\end{pmatrix}\) or \(-\mathbf{j} - \mathbf{k}\)	A1

M1: may see \(3\overrightarrow{OC} = \begin{pmatrix}0\\-3\\-3\end{pmatrix}\); allow one sign error only;
may work with i, j, k notation

A1: do not allow \((0, -1, -1)\);
only penalise coordinate form once

Alternative method 3

Scheme	Marks
\([\overrightarrow{OC} =]\ \frac{2}{3}\begin{pmatrix}-1\\-2\\0\end{pmatrix} + \frac{1}{3}\begin{pmatrix}2\\1\\-3\end{pmatrix}\)	M1
\(\begin{pmatrix}0\\-1\\-1\end{pmatrix}\) or \(-\mathbf{j} - \mathbf{k}\)	A1

M1: ratio theorem; may work with i, j, k notation

A1: do not allow \((0, -1, -1)\);
only penalise coordinate form once