| Exam Board | OCR |
|---|---|
| Module | Further Additional Pure AS (Further Additional Pure AS) |
| Year | 2021 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vector Product and Surfaces |
| Type | Area of triangle using vector product |
| Difficulty | Moderate -0.3 This is a straightforward application of the vector product formula with simple orthogonal unit vectors along coordinate axes. Part (a)(i) requires routine calculation of a×b with very simple arithmetic, part (a)(ii) tests basic geometric understanding of perpendicularity, and part (b) applies the standard formula |a×b|/2 for triangle area. While it's a Further Maths topic (making it slightly harder than typical A-level), the vectors are deliberately simple (axis-aligned) making calculations trivial, placing it slightly below average difficulty overall. |
| Spec | 4.04g Vector product: a x b perpendicular vector8.04a Vector product: definition, magnitude/direction, component form8.04c Areas using vector product: triangles and parallelograms |
| Answer | Marks | Guidance |
|---|---|---|
| 1 | (a) | (i) |
| [1] | 1.1 | |
| (ii) | The VP of a and b must be perpendicular to both; i.e. | |
| in the direction of c | B1 | |
| [1] | 2.4 | Allow “since a, b, c form a right-handed system” |
| Answer | Marks |
|---|---|
| (b) | Any two of |
| Answer | Marks |
|---|---|
| 12 | B1 |
| Answer | Marks |
|---|---|
| [3] | 1.1 |
| Answer | Marks |
|---|---|
| 1.1 | (±) |
Question 1:
1 | (a) | (i) | a × b = (3i) × (4j) = 12(i × j) = 12k = 12c | B1
[1] | 1.1
(ii) | The VP of a and b must be perpendicular to both; i.e.
in the direction of c | B1
[1] | 2.4 | Allow “since a, b, c form a right-handed system”
or “since a, b, c form a set of mutually perpendicular vectors”
(b) | Any two of
b – a = −3i + 4j, c – a = −3i + k , c – b = −4j + k
4
Area ∆ABC = 1 2 AB×AC = 1 2 3 = 1 2 3
12 | B1
M1
A1
[3] | 1.1
1.1
1.1 | (±)
Use of area formula with relevant vectors and attempt at a vector
1
product (condone missing )
2
Correct answer1 The points $A , B$ and $C$ have position vectors $\mathbf { a } = \left( \begin{array} { l } 3 \\ 0 \\ 0 \end{array} \right) , \mathbf { b } = \left( \begin{array} { l } 0 \\ 4 \\ 0 \end{array} \right)$ and $\mathbf { c } = \left( \begin{array} { l } 0 \\ 0 \\ 1 \end{array} \right)$ respectively, relative to the origin $O$.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Calculate $\mathbf { a } \times \mathbf { b }$, giving your answer as a multiple of $\mathbf { c }$.
\item Explain, geometrically, why $\mathbf { a } \times \mathbf { b }$ must be a multiple of $\mathbf { c }$.
\end{enumerate}\item Use a vector product method to calculate the area of triangle $A B C$.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Additional Pure AS 2021 Q1 [5]}}