| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Perpendicular distance point to line |
| Difficulty | Standard +0.3 This is a straightforward multi-part vectors question requiring standard techniques: scalar product for perpendicularity, magnitude calculations for isosceles property, and perpendicular distance formula. All methods are routine A-level procedures with no novel insight required, making it slightly easier than average. |
| Spec | 1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication1.10g Problem solving with vectors: in geometry |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State \(\overrightarrow{AB}\) (or \(\overrightarrow{BA}\)) and \(\overrightarrow{BC}\) (or \(\overrightarrow{CB}\)) in vector form | B1 | |
| Calculate their scalar product | M1 | |
| Show product is zero and confirm angle \(ABC\) is a right angle | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use correct method to calculate the lengths of \(AB\) and \(BC\) | M1 | |
| Show that \(AB = BC\) and the triangle is isosceles | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State a correct equation for the line through \(B\) and \(C\), e.g. \(\mathbf{r} = \mathbf{i}+\mathbf{j}+\mathbf{k}+\lambda(2\mathbf{i}+\mathbf{j}+2\mathbf{k})\) or \(\mathbf{r} = 3\mathbf{i}+2\mathbf{j}+3\mathbf{k}+\mu(-2\mathbf{i}-\mathbf{j}-2\mathbf{k})\) | B1 | |
| Taking a general point of \(BC\) to be \(P\), form an equation in \(\lambda\) by either equating the scalar product of \(\overrightarrow{OP}\) and \(\overrightarrow{BC}\) to zero, or applying Pythagoras to triangle \(OBP\) (or \(OCP\)), or setting the derivative of \( | \overrightarrow{OP} | \) to zero |
| Solve and obtain \(\lambda = -\dfrac{5}{9}\) | A1 | |
| Obtain answer \(\dfrac{1}{3}\sqrt{2}\), or equivalent | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use a scalar product to find the projection \(CN\) (or \(BN\)) of \(OC\) (or \(OB\)) on \(BC\) | M1 | |
| Obtain answer \(CN = \frac{5}{3}\) \(\left(\text{or } BN = \frac{14}{3}\right)\) | A1 | |
| Use Pythagoras to find \(ON\) | M1 | |
| Obtain answer \(\frac{1}{3}\sqrt{2}\), or equivalent | A1 |
## Question 9(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| State $\overrightarrow{AB}$ (or $\overrightarrow{BA}$) and $\overrightarrow{BC}$ (or $\overrightarrow{CB}$) in vector form | B1 | |
| Calculate their scalar product | M1 | |
| Show product is zero and confirm angle $ABC$ is a right angle | A1 | |
---
## Question 9(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use correct method to calculate the lengths of $AB$ and $BC$ | M1 | |
| Show that $AB = BC$ and the triangle is isosceles | A1 | |
---
## Question 9(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| State a correct equation for the line through $B$ and $C$, e.g. $\mathbf{r} = \mathbf{i}+\mathbf{j}+\mathbf{k}+\lambda(2\mathbf{i}+\mathbf{j}+2\mathbf{k})$ or $\mathbf{r} = 3\mathbf{i}+2\mathbf{j}+3\mathbf{k}+\mu(-2\mathbf{i}-\mathbf{j}-2\mathbf{k})$ | B1 | |
| Taking a general point of $BC$ to be $P$, form an equation in $\lambda$ by either equating the scalar product of $\overrightarrow{OP}$ and $\overrightarrow{BC}$ to zero, or applying Pythagoras to triangle $OBP$ (or $OCP$), or setting the derivative of $|\overrightarrow{OP}|$ to zero | M1 | |
| Solve and obtain $\lambda = -\dfrac{5}{9}$ | A1 | |
| Obtain answer $\dfrac{1}{3}\sqrt{2}$, or equivalent | A1 | |
# Question 9(c) [Alternative Method]:
| Answer | Mark | Guidance |
|--------|------|----------|
| Use a scalar product to find the projection $CN$ (or $BN$) of $OC$ (or $OB$) on $BC$ | M1 | |
| Obtain answer $CN = \frac{5}{3}$ $\left(\text{or } BN = \frac{14}{3}\right)$ | A1 | |
| Use Pythagoras to find $ON$ | M1 | |
| Obtain answer $\frac{1}{3}\sqrt{2}$, or equivalent | A1 | |
---9 With respect to the origin $O$, the vertices of a triangle $A B C$ have position vectors
$$\overrightarrow { O A } = 2 \mathbf { i } + 5 \mathbf { k } , \quad \overrightarrow { O B } = 3 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } \quad \text { and } \quad \overrightarrow { O C } = \mathbf { i } + \mathbf { j } + \mathbf { k }$$
\begin{enumerate}[label=(\alph*)]
\item Using a scalar product, show that angle $A B C$ is a right angle.
\item Show that triangle $A B C$ is isosceles.
\item Find the exact length of the perpendicular from $O$ to the line through $B$ and $C$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2020 Q9 [9]}}