Reflection and symmetry - Vectors 3D & Lines

Edexcel C34 2014 January Q10

11 marks Challenging +1.2

10. With respect to a fixed origin $O$, the lines $l _ { 1 }$ and $l _ { 2 }$ are given by the equations $$\begin{aligned} & l _ { 1 } : \mathbf { r } = ( \mathbf { i } + 5 \mathbf { j } + 5 \mathbf { k } ) + \lambda ( 2 \mathbf { i } + \mathbf { j } - \mathbf { k } ) \\ & l _ { 2 } : \mathbf { r } = ( 2 \mathbf { j } + 12 \mathbf { k } ) + \mu ( 3 \mathbf { i } - \mathbf { j } + 5 \mathbf { k } ) \end{aligned}$$ where $\lambda$ and $\mu$ are scalar parameters.

Show that $l _ { 1 }$ and $l _ { 2 }$ meet and find the position vector of their point of intersection.
Show that $l _ { 1 }$ and $l _ { 2 }$ are perpendicular to each other. The point $A$, with position vector $5 \mathbf { i } + 7 \mathbf { j } + 3 \mathbf { k }$, lies on $l _ { 1 }$ The point $B$ is the image of $A$ after reflection in the line $l _ { 2 }$
Find the position vector of $B$. \includegraphics[max width=\textwidth, alt={}, center]{5b698944-41ac-4072-b5e1-c580b7752c39-35_133_163_2604_1786}

Edexcel C4 2008 June Q6

12 marks Challenging +1.2

6. With respect to a fixed origin $O$, the lines $l _ { 1 }$ and $l _ { 2 }$ are given by the equations $$\begin{array} { l l } l _ { 1 } : & \mathbf { r } = ( - 9 \mathbf { i } + 10 \mathbf { k } ) + \lambda ( 2 \mathbf { i } + \mathbf { j } - \mathbf { k } ) \\ l _ { 2 } : & \mathbf { r } = ( 3 \mathbf { i } + \mathbf { j } + 17 \mathbf { k } ) + \mu ( 3 \mathbf { i } - \mathbf { j } + 5 \mathbf { k } ) \end{array}$$ where $\lambda$ and $\mu$ are scalar parameters.

Show that $l _ { 1 }$ and $l _ { 2 }$ meet and find the position vector of their point of intersection.
Show that $l _ { 1 }$ and $l _ { 2 }$ are perpendicular to each other. The point $A$ has position vector $5 \mathbf { i } + 7 \mathbf { j } + 3 \mathbf { k }$.
Show that $A$ lies on $l _ { 1 }$. The point $B$ is the image of $A$ after reflection in the line $l _ { 2 }$.
Find the position vector of $B$.

Edexcel C4 2013 June Q6

12 marks Standard +0.3

6. Relative to a fixed origin $O$, the point $A$ has position vector $21 \mathbf { i } - 17 \mathbf { j } + 6 \mathbf { k }$ and the point $B$ has position vector $25 \mathbf { i } - 14 \mathbf { j } + 18 \mathbf { k }$. The line $l$ has vector equation $$\mathbf { r } = \left( \begin{array} { r } a \\ b \\ 10 \end{array} \right) + \lambda \left( \begin{array} { r } 6 \\ c \\ - 1 \end{array} \right)$$ where $a , b$ and $c$ are constants and $\lambda$ is a parameter.
Given that the point $A$ lies on the line $l$,

find the value of $a$. Given also that the vector $\overrightarrow { A B }$ is perpendicular to $l$,
find the values of $b$ and $c$,
find the distance $A B$. The image of the point $B$ after reflection in the line $l$ is the point $B ^ { \prime }$.
Find the position vector of the point $B ^ { \prime }$. $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ \section*{Question 6 continued} $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$

SPS SPS FM Pure 2025 January Q5

11 marks Standard +0.3

With respect to a fixed origin $O$, the lines $l_1$ and $l_2$ are given by the equations $$l_1: \mathbf{r} = (\mathbf{i} + 5\mathbf{j} + 5\mathbf{k}) + \lambda(2\mathbf{i} + \mathbf{j} - \mathbf{k})$$ $$l_2: \mathbf{r} = (2\mathbf{j} + 12\mathbf{k}) + \mu(3\mathbf{i} - \mathbf{j} + 5\mathbf{k})$$ where $\lambda$ and $\mu$ are scalar parameters.

Show that $l_1$ and $l_2$ meet and find the position vector of their point of intersection. [6]
Show that $l_1$ and $l_2$ are perpendicular to each other. [2]

The point $A$, with position vector $5\mathbf{i} + 7\mathbf{j} + 3\mathbf{k}$, lies on $l_1$ The point $B$ is the image of $A$ after reflection in the line $l_2$

Find the position vector of $B$. [3]

Edexcel AEA 2011 June Q6

19 marks Hard +2.3

The line $L$ has equation $$\mathbf{r} = \begin{pmatrix} 13 \\ -3 \\ -8 \end{pmatrix} + t \begin{pmatrix} -5 \\ 3 \\ 4 \end{pmatrix}$$ The point $P$ has position vector $\begin{pmatrix} -7 \\ 2 \\ 7 \end{pmatrix}$. The point $P'$ is the reflection of $P$ in $L$.

Find the position vector of $P'$. [6]
Show that the point $A$ with position vector $\begin{pmatrix} -7 \\ 9 \\ 8 \end{pmatrix}$ lies on $L$. [1]
Show that angle $PAP' = 120°$. [3]

% Figure 3 shows kite APBP' with angle at A = 120° \includegraphics{figure_3} Figure 3 The point $B$ lies on $L$ and $APBP'$ forms a kite as shown in Figure 3. The area of the kite is $50\sqrt{3}$

Find the position vector of the point $B$. [5]
Show that angle $BPA = 90°$. [2]

The circle $C$ passes through the points $A$, $P$, $P'$ and $B$.

Find the position vector of the centre of $C$. [2]

[Total 19 marks]