| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2014 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Cross Product & Distances |
| Type | Angle between vectors using scalar product |
| Difficulty | Standard +0.3 This is a standard multi-part vectors question requiring routine techniques: finding angle using dot product (part i), using area formula with sine (part ii), and finding a plane equation with constraints (part iii). While it involves multiple steps and careful calculation, all methods are textbook-standard for Further Maths P3 with no novel problem-solving required. Slightly easier than average due to straightforward application of formulas. |
| Spec | 1.10g Problem solving with vectors: in geometry4.04c Scalar product: calculate and use for angles |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Either: State or imply \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\) correctly in component form | B1 | |
| Using the correct processes evaluate the scalar product \(\overrightarrow{AB} \cdot \overrightarrow{AC}\), or equivalent | M1 | |
| Using the correct process for the moduli divide the scalar product by the product of the moduli | M1 | |
| Obtain answer \(\frac{20}{21}\) | A1 | |
| Or: Use correct method to find lengths of all sides of triangle \(ABC\) | M1 | |
| Apply cosine rule correctly to find the cosine of angle \(BAC\) | M1 | |
| Obtain answer \(\frac{20}{21}\) | A1 | |
| Total: 4 marks | ||
| (ii) State an exact value for the sine of angle \(BAC\), e.g. \(\sqrt{41}/21\) | B1√ | |
| Use correct area formula to find the area of triangle \(ABC\) | M1 | |
| Obtain answer \(\frac{1}{2}\sqrt{41}\), or exact equivalent | A1 | |
| [SR: Allow use of a vector product, e.g. \(\overrightarrow{AB} \times \overrightarrow{AC} = -6\mathbf{i} + 2\mathbf{j} - \mathbf{k}\) | B1√. Using correct process for the modulus, divide the modulus by 2 M1. Obtain answer \(\frac{1}{2}\sqrt{41}\) A1.] | Total: 3 marks |
| (iii) Either: State or obtain \(b = 0\) | B1 | |
| Equate scalar product of normal vector and \(\overrightarrow{BC}\) (or \(\overrightarrow{CB}\)) to zero | M1 | |
| Obtain \(a + b - 4c = 0\) (or \(a - 4c = 0\)) | A1 | |
| Substitute a relevant point in \(4x + z = d\) and evaluate \(d\) | M1 | |
| Obtain answer \(4x + z = 9\), or equivalent | A1 | |
| Or1: Attempt to calculate vector product of relevant vectors, e.g. \((\mathbf{i}\mathbf{j}\mathbf{k} - 4\mathbf{k})\) | M1 | |
| Obtain two correct components of the product | A1 | |
| Obtain correct product, e.g. \(-4\mathbf{i} - \mathbf{k}\) | A1 | |
| Substitute a relevant point in \(4x + z = d\) and evaluate \(d\) | M1 | |
| Obtain \(4x + z = 9\), or equivalent | A1 | |
| Or2: Attempt to form a 2-parameter equation for the plane with relevant vectors | M1 | |
| State a correct equation, e.g. \(r = 2\mathbf{i} + 4\mathbf{j} + \mathbf{k} + \lambda(\mathbf{i} + \mathbf{j} - 4\mathbf{k}) + \mu(a\mathbf{i} + \mathbf{j} - 4\mathbf{k})\) | A1 | |
| State 3 equations in \(x, y, z, \lambda\) and \(\mu\) | A1 | |
| Eliminate \(\mu\) | M1 | |
| Obtain answer \(4x + z = 9\), or equivalent | A1 | |
| Or3: State or obtain \(b = 0\) | B1 | |
| Substitute for \(B\) and \(C\) in the plane equation and obtain \(2a + c = d\) and \(3a - 3c = d\) (or \(2a + 4b + c = d\) and \(3a + 5b - 3c = d\)) | B1 | |
| Solve for one ratio, e.g. \(a : d\) | M1 | |
| Obtain \(a : c : d\), or equivalent | M1 | |
| Obtain answer \(4x + z = 9\), or equivalent | A1 | |
| Or4: Attempt to form a determinant equation for the plane with relevant vectors | M1 | |
| State a correct equation, e.g. \(\begin{vmatrix} x - 2 & y - 4 & z - 1 \\ 0 & 1 & 0 \\ 1 & 1 & -4 \end{vmatrix} = 0\) | A1 | |
| Attempt to use a correct method to expand the determinant | M1 | |
| Obtain two correct terms of a 3-term expansion, or equivalent | A1 | |
| Obtain answer \(4x + z = 9\), or equivalent | A1 | |
| Total: 5 marks |
**(i)** Either: State or imply $\overrightarrow{AB}$ and $\overrightarrow{AC}$ correctly in component form | B1 |
Using the correct processes evaluate the scalar product $\overrightarrow{AB} \cdot \overrightarrow{AC}$, or equivalent | M1 |
Using the correct process for the moduli divide the scalar product by the product of the moduli | M1 |
Obtain answer $\frac{20}{21}$ | A1 |
Or: Use correct method to find lengths of all sides of triangle $ABC$ | M1 |
Apply cosine rule correctly to find the cosine of angle $BAC$ | M1 |
Obtain answer $\frac{20}{21}$ | A1 |
| Total: 4 marks |
**(ii)** State an exact value for the sine of angle $BAC$, e.g. $\sqrt{41}/21$ | B1√ |
Use correct area formula to find the area of triangle $ABC$ | M1 |
Obtain answer $\frac{1}{2}\sqrt{41}$, or exact equivalent | A1 |
[SR: Allow use of a vector product, e.g. $\overrightarrow{AB} \times \overrightarrow{AC} = -6\mathbf{i} + 2\mathbf{j} - \mathbf{k}$ | B1√. Using correct process for the modulus, divide the modulus by 2 M1. Obtain answer $\frac{1}{2}\sqrt{41}$ A1.] | Total: 3 marks |
**(iii)** Either: State or obtain $b = 0$ | B1 |
Equate scalar product of normal vector and $\overrightarrow{BC}$ (or $\overrightarrow{CB}$) to zero | M1 |
Obtain $a + b - 4c = 0$ (or $a - 4c = 0$) | A1 |
Substitute a relevant point in $4x + z = d$ and evaluate $d$ | M1 |
Obtain answer $4x + z = 9$, or equivalent | A1 |
Or1: Attempt to calculate vector product of relevant vectors, e.g. $(\mathbf{i}\mathbf{j}\mathbf{k} - 4\mathbf{k})$ | M1 |
Obtain two correct components of the product | A1 |
Obtain correct product, e.g. $-4\mathbf{i} - \mathbf{k}$ | A1 |
Substitute a relevant point in $4x + z = d$ and evaluate $d$ | M1 |
Obtain $4x + z = 9$, or equivalent | A1 |
Or2: Attempt to form a 2-parameter equation for the plane with relevant vectors | M1 |
State a correct equation, e.g. $r = 2\mathbf{i} + 4\mathbf{j} + \mathbf{k} + \lambda(\mathbf{i} + \mathbf{j} - 4\mathbf{k}) + \mu(a\mathbf{i} + \mathbf{j} - 4\mathbf{k})$ | A1 |
State 3 equations in $x, y, z, \lambda$ and $\mu$ | A1 |
Eliminate $\mu$ | M1 |
Obtain answer $4x + z = 9$, or equivalent | A1 |
Or3: State or obtain $b = 0$ | B1 |
Substitute for $B$ and $C$ in the plane equation and obtain $2a + c = d$ and $3a - 3c = d$ (or $2a + 4b + c = d$ and $3a + 5b - 3c = d$) | B1 |
Solve for one ratio, e.g. $a : d$ | M1 |
Obtain $a : c : d$, or equivalent | M1 |
Obtain answer $4x + z = 9$, or equivalent | A1 |
Or4: Attempt to form a determinant equation for the plane with relevant vectors | M1 |
State a correct equation, e.g. $\begin{vmatrix} x - 2 & y - 4 & z - 1 \\ 0 & 1 & 0 \\ 1 & 1 & -4 \end{vmatrix} = 0$ | A1 |
Attempt to use a correct method to expand the determinant | M1 |
Obtain two correct terms of a 3-term expansion, or equivalent | A1 |
Obtain answer $4x + z = 9$, or equivalent | A1 |
| Total: 5 marks |Referred to the origin $O$, the points $A$, $B$ and $C$ have position vectors given by
$$\overrightarrow{OA} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k}, \quad \overrightarrow{OB} = 2\mathbf{i} + 4\mathbf{j} + \mathbf{k} \quad \text{and} \quad \overrightarrow{OC} = 3\mathbf{i} + 5\mathbf{j} - 3\mathbf{k}.$$
\begin{enumerate}[label=(\roman*)]
\item Find the exact value of the cosine of angle $BAC$. [4]
\item Hence find the exact value of the area of triangle $ABC$. [3]
\item Find the equation of the plane which is parallel to the $y$-axis and contains the line through $B$ and $C$. Give your answer in the form $ax + by + cz = d$. [5]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2014 Q10 [12]}}