Triangle and parallelogram problems

CAIE P1 2013 June Q8

8 marks Moderate -0.3

8 \includegraphics[max width=\textwidth, alt={}, center]{13cfb59a-7781-4786-a625-919b01a2a4f0-3_716_437_1137_854} The diagram shows a parallelogram $O A B C$ in which $$\overrightarrow { O A } = \left( \begin{array} { r } 3 \\ 3 \\ - 4 \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { l } 5 \\ 0 \\ 2 \end{array} \right)$$

Use a scalar product to find angle $B O C$.
Find a vector which has magnitude 35 and is parallel to the vector $\overrightarrow { O C }$.

CAIE P1 2017 November Q9

9 marks Standard +0.3

9 \includegraphics[max width=\textwidth, alt={}, center]{518bb805-5b14-4b41-94fd-38a31a90c218-16_533_601_258_772} The diagram shows a trapezium $O A B C$ in which $O A$ is parallel to $C B$. The position vectors of $A$ and $B$ relative to the origin $O$ are given by $\overrightarrow { O A } = \left( \begin{array} { r } 2 \\ - 2 \\ - 1 \end{array} \right)$ and $\overrightarrow { O B } = \left( \begin{array} { l } 6 \\ 1 \\ 1 \end{array} \right)$.

Show that angle $O A B$ is $90 ^ { \circ }$.
The magnitude of $\overrightarrow { C B }$ is three times the magnitude of $\overrightarrow { O A }$.
Find the position vector of $C$.
Find the exact area of the trapezium $O A B C$, giving your answer in the form $a \sqrt { } b$, where $a$ and $b$ are integers.

OCR C4 2013 January Q8

7 marks Standard +0.3

8 The points $A ( 3,2,1 ) , B ( 5,4 , - 3 ) , C ( 3,17 , - 4 )$ and $D ( 1,6,3 )$ form a quadrilateral $A B C D$.

Show that $A B = A D$.
Find a vector equation of the line through $A$ and the mid-point of $B D$.
Show that $C$ lies on the line found in part (ii).
What type of quadrilateral is $A B C D$ ?

Edexcel Paper 1 Specimen Q4

5 marks Moderate -0.8

Relative to a fixed origin $O$,
the point $A$ has position vector $\mathbf { i } + 7 \mathbf { j } - 2 \mathbf { k }$,
the point $B$ has position vector $4 \mathbf { i } + 3 \mathbf { j } + 3 \mathbf { k }$,
and the point $C$ has position vector $2 \mathbf { i } + 10 \mathbf { j } + 9 \mathbf { k }$.
Given that $A B C D$ is a parallelogram,
1. find the position vector of point $D$.
The vector $\overrightarrow { A X }$ has the same direction as $\overrightarrow { A B }$.
Given that $| \overrightarrow { A X } | = 10 \sqrt { 2 }$,
find the position vector of $X$.

AQA C4 2011 January Q8

14 marks Standard +0.3

8 The coordinates of the points $A$ and $B$ are $( 3 , - 2,4 )$ and $( 6,0,3 )$ respectively.
The line $l _ { 1 }$ has equation $\mathbf { r } = \left[ \begin{array} { r } 3 \\ - 2 \\ 4 \end{array} \right] + \lambda \left[ \begin{array} { r } 2 \\ - 1 \\ 3 \end{array} \right]$.

1. Find the vector $\overrightarrow { A B }$.
2. Calculate the acute angle between $\overrightarrow { A B }$ and the line $l _ { 1 }$, giving your answer to the nearest $0.1 ^ { \circ }$.
The point $D$ lies on $l _ { 1 }$ where $\lambda = 2$. The line $l _ { 2 }$ passes through $D$ and is parallel to $A B$.
1. Find a vector equation of line $l _ { 2 }$ with parameter $\mu$.
2. The diagram shows a symmetrical trapezium $A B C D$, with angle $D A B$ equal to angle $A B C$. \includegraphics[max width=\textwidth, alt={}, center]{5fe2527a-33da-4076-b3fa-4cab545336ec-9_620_675_1197_726} The point $C$ lies on line $l _ { 2 }$. The length of $A D$ is equal to the length of $B C$. Find the coordinates of $C$.

AQA C4 2012 January Q8

12 marks Standard +0.3

8 The points $A$ and $B$ have coordinates $( 4 , - 2,3 )$ and $( 2,0 , - 1 )$ respectively. The line $l$ passes through $A$ and has equation $\mathbf { r } = \left[ \begin{array} { r } 4 \\ - 2 \\ 3 \end{array} \right] + \lambda \left[ \begin{array} { r } 1 \\ 5 \\ - 2 \end{array} \right]$.

1. Find the vector $\overrightarrow { A B }$.
2. Find the acute angle between $A B$ and the line $l$, giving your answer to the nearest degree.
The point $C$ lies on the line $l$ such that the angle $A B C$ is a right angle. Given that $A B C D$ is a rectangle, find the coordinates of the point $D$.

AQA C4 2013 January Q6

15 marks Standard +0.3

6

The points $A , B$ and $C$ have coordinates $( 3,1 , - 6 ) , ( 5 , - 2,0 )$ and $( 8 , - 4 , - 6 )$ respectively.
1. Show that the vector $\overrightarrow { A C }$ is given by $\overrightarrow { A C } = n \left[ \begin{array} { r } 1 \\ - 1 \\ 0 \end{array} \right]$, where $n$ is an integer.
2. Show that the acute angle $A C B$ is given by $\cos ^ { - 1 } \left( \frac { 5 \sqrt { 2 } } { 14 } \right)$.
Find a vector equation of the line $A C$.
The point $D$ has coordinates $( 6 , - 1 , p )$. It is given that the lines $A C$ and $B D$ intersect.
1. Find the value of $p$.
2. Show that $A B C D$ is a rhombus, and state the length of each of its sides.

AQA C4 2007 January Q6

13 marks Moderate -0.3

6 The points $A , B$ and $C$ have coordinates $( 3 , - 2,4 ) , ( 5,4,0 )$ and $( 11,6 , - 4 )$ respectively.

1. Find the vector $\overrightarrow { B A }$.
2. Show that the size of angle $A B C$ is $\cos ^ { - 1 } \left( - \frac { 5 } { 7 } \right)$.
The line $l$ has equation $\mathbf { r } = \left[ \begin{array} { r } 8 \\ - 3 \\ 2 \end{array} \right] + \lambda \left[ \begin{array} { r } 1 \\ 3 \\ - 2 \end{array} \right]$.
1. Verify that $C$ lies on $l$.
2. Show that $A B$ is parallel to $l$.
The quadrilateral $A B C D$ is a parallelogram. Find the coordinates of $D$.

AQA C4 2009 June Q7

10 marks Moderate -0.3

7 The points $A$ and $B$ have coordinates ( $3 , - 2,5$ ) and ( $4,0,1$ ) respectively. The line $l _ { 1 }$ has equation $\mathbf { r } = \left[ \begin{array} { r } 6 \\ - 1 \\ 5 \end{array} \right] + \lambda \left[ \begin{array} { r } 2 \\ - 1 \\ 4 \end{array} \right]$.

Find the distance between the points $A$ and $B$.
Verify that $B$ lies on $l _ { 1 }$.
(2 marks)
The line $l _ { 2 }$ passes through $A$ and has equation $\mathbf { r } = \left[ \begin{array} { r } 3 \\ - 2 \\ 5 \end{array} \right] + \mu \left[ \begin{array} { r } - 1 \\ 3 \\ - 8 \end{array} \right]$. The lines $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $C$. Show that the points $A , B$ and $C$ form an isosceles triangle.
(6 marks)

CAIE P3 2024 November Q9

11 marks Standard +0.3

With respect to the origin $O$, the points $A$, $B$ and $C$ have position vectors given by $$\overrightarrow{OA} = \begin{pmatrix} 2 \\ 1 \\ -3 \end{pmatrix}, \quad \overrightarrow{OB} = \begin{pmatrix} 0 \\ 4 \\ 1 \end{pmatrix} \quad \text{and} \quad \overrightarrow{OC} = \begin{pmatrix} -3 \\ -2 \\ 2 \end{pmatrix}.$$

The point $D$ is such that $ABCD$ is a trapezium with $\overrightarrow{DC} = 3\overrightarrow{AB}$. Find the position vector of $D$. [2]
The diagonals of the trapezium intersect at the point $P$. Find the position vector of $P$. [5]
Using a scalar product, calculate angle $ABC$. [4]

OCR C4 2005 June Q5

7 marks Moderate -0.3

$ABCD$ is a parallelogram. The position vectors of $A$, $B$ and $C$ are given respectively by $$\mathbf{a} = 2\mathbf{i} + \mathbf{j} + 3\mathbf{k}, \quad \mathbf{b} = 3\mathbf{i} - 2\mathbf{j}, \quad \mathbf{c} = \mathbf{i} - \mathbf{j} - 2\mathbf{k}.$$

Find the position vector of $D$. [3]
Determine, to the nearest degree, the angle $ABC$. [4]

AQA Paper 2 2018 June Q14

6 marks Moderate -0.8

A quadrilateral has vertices A, B, C and D with position vectors given by $$\overrightarrow{OA} = \begin{pmatrix} 3 \\ 5 \\ 1 \end{pmatrix}, \overrightarrow{OB} = \begin{pmatrix} -1 \\ 2 \\ 7 \end{pmatrix}, \overrightarrow{OC} = \begin{pmatrix} 0 \\ 7 \\ 6 \end{pmatrix} \text{ and } \overrightarrow{OD} = \begin{pmatrix} 4 \\ 10 \\ 0 \end{pmatrix}$$

Write down the vector $\overrightarrow{AB}$ [1 mark]
Show that ABCD is a parallelogram, but not a rhombus. [5 marks]