| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2023 |
| Session | March |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Cross Product & Distances |
| Type | Volume of tetrahedron using scalar triple product |
| Difficulty | Standard +0.8 This is a Further Maths question requiring knowledge of the scalar triple product formula for tetrahedron volume (V = 1/6|a·(b×c)|), involving vector subtraction to find edges, cross product computation, dot product, and absolute value. While the calculation is systematic, it's beyond standard A-level and requires multiple vector operations with careful arithmetic—moderately challenging for Further Maths students. |
| Spec | 1.10b Vectors in 3D: i,j,k notation1.10c Magnitude and direction: of vectors4.04a Line equations: 2D and 3D, cartesian and vector forms4.04e Line intersections: parallel, skew, or intersecting |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Carry out correct process for evaluating scalar product of \(\overrightarrow{OA}\) and \(\overrightarrow{OB}\) | M1 | \(\pm(3,-1,2)\cdot(1,2,-3) = \pm(3-2-6) = [-5]\) |
| Using correct process for moduli, divide scalar product by product of moduli and obtain \(\cos^{-1}\left\{\frac{\pm(3-2-6)}{\sqrt{3^2+(-1)^2+2^2}\,\sqrt{1^2+2^2+(-3)^2}}\right\}\) | A1 | |
| Obtain answer \(110.9°\) or \(1.94^c\) | A1 | |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use a correct method to form equation for line through \(AB\) | M1 | |
| Obtain \(\mathbf{r} = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k} + \mu_1(2\mathbf{i} - 3\mathbf{j} + 5\mathbf{k})\) | A1 | OE e.g. \(\mathbf{r} = \mathbf{i}+2\mathbf{j}-3\mathbf{k}+\mu_2(-2\mathbf{i}+3\mathbf{j}-5\mathbf{k})\). Need r or \((x,y,z)\) |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Obtain correct equation for line through \(CD\): e.g. \([\mathbf{r}=]\,\mathbf{i}-2\mathbf{j}+5\mathbf{k}+\lambda_1(-4\mathbf{i}+4\mathbf{j}-6\mathbf{k})\) | B1 | OE e.g. \([\mathbf{r}=]\,5\mathbf{i}-6\mathbf{j}+11\mathbf{k}+\lambda_2(-4\mathbf{i}+4\mathbf{j}-6\mathbf{k})\). r can be omitted or another symbol used |
| Equate two pairs of components of general points on line \(l\) and line \(CD\) and solve for \(\lambda\) or \(\mu\) | M1 | |
| Obtain e.g. \(\lambda_1=-2\) or \(\mu_1=3\) or \(\lambda_2=-1\) or \(\mu_2=-4\) | A1 | |
| Obtain position vector \(9\mathbf{i}-10\mathbf{j}+17\mathbf{k}\) | A1 | Condone \((9,-10,17)\) but not \((9\mathbf{i},-10\mathbf{j},17\mathbf{k})\) |
| Total: 4 |
## Question 10(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Carry out correct process for evaluating scalar product of $\overrightarrow{OA}$ and $\overrightarrow{OB}$ | M1 | $\pm(3,-1,2)\cdot(1,2,-3) = \pm(3-2-6) = [-5]$ |
| Using correct process for moduli, divide scalar product by product of moduli and obtain $\cos^{-1}\left\{\frac{\pm(3-2-6)}{\sqrt{3^2+(-1)^2+2^2}\,\sqrt{1^2+2^2+(-3)^2}}\right\}$ | A1 | |
| Obtain answer $110.9°$ or $1.94^c$ | A1 | |
| **Total: 3** | | |
## Question 10(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use a correct method to form equation for line through $AB$ | M1 | |
| Obtain $\mathbf{r} = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k} + \mu_1(2\mathbf{i} - 3\mathbf{j} + 5\mathbf{k})$ | A1 | OE e.g. $\mathbf{r} = \mathbf{i}+2\mathbf{j}-3\mathbf{k}+\mu_2(-2\mathbf{i}+3\mathbf{j}-5\mathbf{k})$. Need **r** or $(x,y,z)$ |
| **Total: 2** | | |
## Question 10(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Obtain correct equation for line through $CD$: e.g. $[\mathbf{r}=]\,\mathbf{i}-2\mathbf{j}+5\mathbf{k}+\lambda_1(-4\mathbf{i}+4\mathbf{j}-6\mathbf{k})$ | B1 | OE e.g. $[\mathbf{r}=]\,5\mathbf{i}-6\mathbf{j}+11\mathbf{k}+\lambda_2(-4\mathbf{i}+4\mathbf{j}-6\mathbf{k})$. **r** can be omitted or another symbol used |
| Equate two pairs of components of general points on line $l$ and line $CD$ and solve for $\lambda$ or $\mu$ | M1 | |
| Obtain e.g. $\lambda_1=-2$ or $\mu_1=3$ or $\lambda_2=-1$ or $\mu_2=-4$ | A1 | |
| Obtain position vector $9\mathbf{i}-10\mathbf{j}+17\mathbf{k}$ | A1 | Condone $(9,-10,17)$ but not $(9\mathbf{i},-10\mathbf{j},17\mathbf{k})$ |
| **Total: 4** | | |10 With respect to the origin $O$, the points $A , B , C$ and $D$ have position vectors given by
$$\overrightarrow { O A } = \left( \begin{array} { r }
3 \\
- 1 \\
2
\end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r }
1 \\
2 \\
- 3
\end{array} \right) , \quad \overrightarrow { O C } = \left( \begin{array} { r }
1 \\
- 2 \\
5
\end{array} \right) \quad \text { and } \quad \overrightarrow { O D } = \left( \begin{array} { r }
5 \\
- 6 \\
11
\end{array} \right)$$
\begin{enumerate}[label=(\alph*)]
\item Find the obtuse angle between the vectors $\overrightarrow { O A }$ and $\overrightarrow { O B }$.\\
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The line $l$ passes through the points $A$ and $B$.
\item Find a vector equation for the line $l$.\\
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\item Find the position vector of the point of intersection of the line $l$ and the line passing through $C$ and $D$.\\
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\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2023 Q10 [9]}}