Geometric properties using vectors - Vectors Introduction & 2D

Edexcel P1 2023 January Q2

5 marks Moderate -0.8

The points $P , Q$ and $R$ have coordinates (-3, 7), (9, 11) and (12, 2) respectively.
1. Prove that angle $P Q R = 90 ^ { \circ }$
Given that the point $S$ is such that $P Q R S$ forms a rectangle,
find the coordinates of $S$.

Edexcel C4 2007 January Q7

14 marks Standard +0.3

7. The point $A$ has position vector $\mathbf { a } = 2 \mathbf { i } + 2 \mathbf { j } + \mathbf { k }$ and the point $B$ has position vector $\mathbf { b } = \mathbf { i } + \mathbf { j } - 4 \mathbf { k }$, relative to an origin $O$.

Find the position vector of the point $C$, with position vector $\mathbf { c }$, given by $$\mathbf { c } = \mathbf { a } + \mathbf { b } .$$
Show that $O A C B$ is a rectangle, and find its exact area. The diagonals of the rectangle, $A B$ and $O C$, meet at the point $D$.
Write down the position vector of the point $D$.
Find the size of the angle $A D C$.

OCR C4 Specimen Q4

7 marks Moderate -0.5

4 \includegraphics[max width=\textwidth, alt={}, center]{798da17d-0af5-4aa6-b731-564642dc28d5-2_428_572_861_760} As shown in the diagram the points $A$ and $B$ have position vectors $\mathbf { a }$ and $\mathbf { b }$ with respect to the origin $O$.

Make a sketch of the diagram, and mark the points $C , D$ and $E$ such that $\overrightarrow { O C } = 2 \mathbf { a } , \overrightarrow { O D } = 2 \mathbf { a } + \mathbf { b }$ and $\overrightarrow { O E } = \frac { 1 } { 3 } \overrightarrow { O D }$.
By expressing suitable vectors in terms of $\mathbf { a }$ and $\mathbf { b }$, prove that $E$ lies on the line joining $A$ and $B$.

Edexcel AEA 2003 June Q1

5 marks Challenging +1.2

1. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{25f0c7cc-0701-4836-931e-0eff5145e029-2_433_549_270_773}

\end{figure} The point $A$ is a distance 1 unit from the fixed origin $O$ .Its position vector is $\mathbf { a } = \frac { 1 } { \sqrt { 2 } } ( \mathbf { i } + \mathbf { j } )$ . The point $B$ has position vector $\mathbf { a } + \mathbf { j }$ ,as shown in Figure 1. By considering $\triangle O A B$ ,prove that $\tan \frac { 3 \pi } { 8 } = 1 + \sqrt { } 2$ .

Edexcel AEA 2017 Specimen Q3

12 marks Challenging +1.8

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{05b21c5d-5958-4267-b1e6-3d1ed20d5609-08_609_631_264_724} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure}

\includegraphics[max width=\textwidth, alt={}]{05b21c5d-5958-4267-b1e6-3d1ed20d5609-08_172_168_781_1548}

Figure 1 shows a regular pentagon $O A B C D$. The vectors $\mathbf { p }$ and $\mathbf { q }$ are defined by $\mathbf { p } = \overrightarrow { O A }$ and $\mathbf { q } = \overrightarrow { O D }$ respectively. Let $k$ be the number such that $\overrightarrow { D B } = k \overrightarrow { O A }$.

Write down $\overrightarrow { A C }$ in terms of $\mathbf { p } , \mathbf { q }$ and $k$ as appropriate.
Show that $\overrightarrow { C D } = - \mathbf { p } - \frac { 1 } { k } \mathbf { q }$
Hence find the value of $k$ By considering triangle $D B C$, or otherwise,
find the exact value of $\sin 54 ^ { \circ }$

Edexcel AS Paper 1 2024 June Q3

8 marks Moderate -0.8

Relative to a fixed origin $O$,

point $P$ has position vector $9 \mathbf { i } - 8 \mathbf { j }$
point $Q$ has position vector $3 \mathbf { i } - 5 \mathbf { j }$
1. Find $\overrightarrow { P Q }$

Given that $R$ is the point such that $\overrightarrow { Q R } = 9 \mathbf { i } + 18 \mathbf { j }$

show that angle $P Q R = 90 ^ { \circ }$ Given also that $S$ is the point such that $\overrightarrow { P S } = 3 \overrightarrow { Q R }$

find the exact area of $P Q R S$

Edexcel AS Paper 1 Specimen Q2

4 marks Moderate -0.8

The quadrilateral $O A B C$ has $\overrightarrow { O A } = 4 \mathbf { i } + 2 \mathbf { j } , \overrightarrow { O B } = 6 \mathbf { i } - 3 \mathbf { j }$ and $\overrightarrow { O C } = 8 \mathbf { i } - 20 \mathbf { j }$.
1. Find $\overrightarrow { A B }$.
2. Show that quadrilateral $O A B C$ is a trapezium.

Edexcel Paper 2 2023 June Q6

6 marks Moderate -0.8

Relative to a fixed origin $O$,

$A$ is the point with position vector $12 \mathbf { i }$
$B$ is the point with position vector $16 \mathbf { j }$
$C$ is the point with position vector $( 50 \mathbf { i } + 136 \mathbf { j } )$
$D$ is the point with position vector $( 22 \mathbf { i } + 24 \mathbf { j } )$
1. Show that $A D$ is parallel to $B C$.

Points $A , B , C$ and $D$ are used to model the vertices of a running track in the shape of a quadrilateral. Runners complete one lap by running along all four sides of the track.
The lengths of the sides are measured in metres. Given that a particular runner takes exactly 5 minutes to complete 2 laps,

calculate the average speed of this runner, giving the answer in kilometres per hour.

OCR PURE Q2

5 marks Moderate -0.3

2 Points $A$ and $B$ have position vectors $\binom { - 3 } { 4 }$ and $\binom { 1 } { 2 }$ respectively.
Point $C$ has position vector $\binom { p } { 1 }$ and $A B C$ is a straight line.

Find $p$. Point $D$ has position vector $\binom { q } { 1 }$ and angle $A B D = 90 ^ { \circ }$.
Determine the value of $q$.

OCR MEI AS Paper 1 2020 November Q3

5 marks Moderate -0.8

3 Fig. 3 shows a triangle PQR . The vector $\overrightarrow { \mathrm { PQ } }$ is $\mathbf { i } + 7 \mathbf { j }$ and the vector $\overrightarrow { \mathrm { QR } }$ is $4 \mathbf { i } - 12 \mathbf { j }$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a1b6c827-7d74-4527-9b60-58872e3d5ef7-3_412_234_1736_244} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure}

Show that the triangle PQR is isosceles. The point P has position vector $- 3 \mathbf { i } - \mathbf { j }$. The point S is added so that PQRS is a parallelogram.
Find the position vector of S .

OCR MEI Paper 3 2023 June Q6

10 marks Standard +0.3

6

Quadrilateral KLMN has vertices $\mathrm { K } ( - 4,1 ) , \mathrm { L } ( 5 , - 1 ) , \mathrm { M } ( 6,2 )$ and $\mathrm { N } ( 2,5 )$, as shown in Fig. 6.1. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Fig. 6.1} \includegraphics[alt={},max width=\textwidth]{20639e13-01cc-4d96-b694-fb3cf1828f4d-06_567_1004_404_319}
\end{figure}
1. Find the coordinates of the following midpoints.
  - P , the midpoint of KL
  - Q, the midpoint of LM
  - R, the midpoint of MN
  - S, the midpoint of NK
  - Verify that PQRS is a parallelogram.
  - TVWX is a quadrilateral as shown in Fig. 6.2.
  Points A and B divide side TV into 3 equal parts. Points C and D divide side VW into 3 equal parts. Points E and F divide side WX into 3 equal parts. Points G and H divide side TX into 3 equal parts. $\overrightarrow { \mathrm { TA } } = \mathbf { a } , \quad \overrightarrow { \mathrm { TH } } = \mathbf { b } , \quad \overrightarrow { \mathrm { VC } } = \mathbf { c }$. \begin{figure}[h]
  \captionsetup{labelformat=empty} \caption{Fig. 6.2} \includegraphics[alt={},max width=\textwidth]{20639e13-01cc-4d96-b694-fb3cf1828f4d-06_577_671_1877_319}
  \end{figure}
  1. Show that $\overrightarrow { \mathrm { WX } } = k ( - \mathbf { a } + \mathbf { b } - \mathbf { c } )$, where $k$ is a constant to be determined.
  2. Verify that AH is parallel to DE .
  3. Verify that BC is parallel to GF .

OCR MEI M1 Q3

8 marks Moderate -0.8

3 The vectors $\mathbf { p }$ and $\mathbf { q }$ are given by $$\mathbf { p } = 8 \mathbf { i } + \mathbf { j } \text { and } \mathbf { q } = 4 \mathbf { i } - 7 \mathbf { j } .$$

Show that $\mathbf { p }$ and $\mathbf { q }$ are equal in magnitude.
Show that $\mathbf { p } + \mathbf { q }$ is parallel to $2 \mathbf { i } - \mathbf { j }$.
Draw $\mathbf { p } + \mathbf { q }$ and $\mathbf { p } - \mathbf { q }$ on the grid. Write down the angle between these two vectors.

AQA Paper 2 2019 June Q15

9 marks Moderate -0.3

Four buoys on the surface of a large, calm lake are located at $A$, $B$, $C$ and $D$ with position vectors given by $$\overrightarrow{OA} = \begin{bmatrix} 410 \\ 710 \end{bmatrix}, \overrightarrow{OB} = \begin{bmatrix} -210 \\ 530 \end{bmatrix}, \overrightarrow{OC} = \begin{bmatrix} -340 \\ -310 \end{bmatrix} \text{ and } \overrightarrow{OD} = \begin{bmatrix} 590 \\ -40 \end{bmatrix}$$ All values are in metres.

Prove that the quadrilateral $ABCD$ is a trapezium but not a parallelogram. [5 marks]
A speed boat travels directly from $B$ to $C$ at a constant speed in 50 seconds. Find the speed of the boat between $B$ and $C$. [4 marks]

OCR PURE Q7

7 marks Moderate -0.8

$OABC$ is a parallelogram with $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OC} = \mathbf{c}$. $P$ is the midpoint of $AC$. \includegraphics{figure_7}

Find the following in terms of $\mathbf{a}$ and $\mathbf{c}$, simplifying your answers.
1. $\overrightarrow{AC}$ [1]
2. $\overrightarrow{OP}$ [2]
Hence prove that the diagonals of a parallelogram bisect one another. [4]

OCR PURE Q2

8 marks Moderate -0.3

The points $A$, $B$ and $C$ have position vectors $\begin{pmatrix} -4 \\ 3 \end{pmatrix}$, $\begin{pmatrix} -3 \\ 6 \end{pmatrix}$ and $\begin{pmatrix} -1 \\ 12 \end{pmatrix}$ respectively.
1. Show that $B$ lies on $AC$. [2]
2. Find the ratio $AB : BC$. [1]
The diagram shows the line $x + y = 6$ and the point $P(2, 4)$ that lies on the line. A copy of the diagram is given in the Printed Answer Booklet. \includegraphics{figure_1} The distinct point $Q$ also lies on the line $x + y = 6$ and is such that $|\overrightarrow{OQ}| = |\overrightarrow{OP}|$, where $O$ is the origin. Find the magnitude and direction of the vector $\overrightarrow{PQ}$. [3]
The point $R$ is such that $\overrightarrow{PR}$ is perpendicular to $\overrightarrow{OP}$ and $|\overrightarrow{PR}| = \frac{1}{2}|\overrightarrow{OP}|$. Write down the position vectors of the two possible positions of the point $R$. [2]

WJEC Unit 1 2019 June Q06

5 marks Moderate -0.8

$OABC$ is a parallelogram with $O$ as origin. \includegraphics{figure_6} The position vector of $A$ is $\mathbf{a}$ and the position vector of $C$ is $\mathbf{c}$. The midpoint of $AB$ is $D$. The point $E$ divides the line $CB$ such that $CE : EB = 2 : 1$.

Find, in terms of $\mathbf{a}$ and $\mathbf{c}$,
1. the vector $\overrightarrow{AC}$,
2. the position vector of $D$,
3. the position vector of $E$. [3]
Determine whether or not $\overrightarrow{DE}$ is parallel to $\overrightarrow{AC}$, clearly stating your reason. [2]

SPS SPS SM Pure 2023 September Q5

7 marks Moderate -0.8

Relative to the origin $O$, the points $A$, $B$ and $C$ have position vectors $4\mathbf{i} + 2\mathbf{j}$, $3\mathbf{i} + 4\mathbf{j}$ and $-\mathbf{i} + 12\mathbf{j}$, respectively.

Find the magnitude of the vector $\overrightarrow{OC}$ [2]
Find the angle that the vector $\overrightarrow{OB}$ makes with the vector $\mathbf{j}$ to the nearest degree [2]
Show that the points $A$, $B$ and $C$ are collinear [3]

OCR H240/01 2017 Specimen Q2

5 marks Moderate -0.8

The points A, B and C have position vectors $\mathbf{3i - 4j + 2k}$, $\mathbf{-i + 6k}$ and $\mathbf{7i - 4j - 2k}$ respectively. M is the midpoint of BC.

Show that the magnitude of $\overrightarrow{OM}$ is equal to $\sqrt{17}$. [2]

Point D is such that $\overrightarrow{BC} = \overrightarrow{AD}$.

Show that position vector of the point D is $\mathbf{1i - 8j - 6k}$. [3]