Reflection in plane - Vectors: Lines & Planes

CAIE P3 2024 June Q9

9 marks Challenging +1.2

9 The equations of two straight lines $l _ { 1 }$ and $l _ { 2 }$ are $$l _ { 1 } : \quad \mathbf { r } = \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } + \lambda ( 2 \mathbf { i } - \mathbf { j } + a \mathbf { k } ) \quad \text { and } \quad l _ { 2 } : \quad \mathbf { r } = - \mathbf { i } - \mathbf { j } - \mathbf { k } + \mu ( 3 \mathbf { i } - 2 \mathbf { j } - 2 \mathbf { k } ) ,$$ where $a$ is a constant.
The lines $l _ { 1 }$ and $l _ { 2 }$ are perpendicular.

Show that $a = 4$.
The lines $l _ { 1 }$ and $l _ { 2 }$ also intersect.
Find the position vector of the point of intersection.
The point $A$ has position vector $- 5 \mathbf { i } + \mathbf { j } - 9 \mathbf { k }$.
Show that $A$ lies on $l _ { 1 }$.
The point $B$ is the image of $A$ after a reflection in the line $l _ { 2 }$.
Find the position vector of $B$.

Edexcel P4 2023 June Q4

10 marks Standard +0.8

Relative to a fixed origin $O$,

the point $A$ has position vector $4 \mathbf { i } + 8 \mathbf { j } + \mathbf { k }$
the point $B$ has position vector $5 \mathbf { i } + 6 \mathbf { j } + 3 \mathbf { k }$
the point $P$ has position vector $2 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }$

The straight line $l$ passes through $A$ and $B$.

Find a vector equation for $l$. The point $C$ lies on $l$ so that $P C$ is perpendicular to $l$.
Find the coordinates of $C$. The point $P ^ { \prime }$ is the reflection of $P$ in the line $l$.
Find the coordinates of $P ^ { \prime }$
Hence find $\left| \overrightarrow { P P ^ { \prime } } \right|$, giving your answer as a simplified surd.

Edexcel F3 2014 June Q8

13 marks Challenging +1.2

8. The line $l$ has equation $$\mathbf { r } = ( 2 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } ) + \lambda ( 3 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } ) , \text { where } \lambda \text { is a scalar parameter, }$$ and the plane $\Pi$ has equation $$\mathbf { r } . ( \mathbf { i } + \mathbf { j } - 2 \mathbf { k } ) = 19$$

Find the coordinates of the point of intersection of $l$ and $\Pi$. The perpendicular to $\Pi$ from the point $A ( 2,1 , - 2 )$ meets $\Pi$ at the point $B$.
Verify that the coordinates of $B$ are $( 4,3 , - 6 )$. The point $A ( 2,1 , - 2 )$ is reflected in the plane $\Pi$ to give the image point $A ^ { \prime }$.
Find the coordinates of the point $A ^ { \prime }$.
Find an equation for the line obtained by reflecting the line $l$ in the plane $\Pi$, giving your answer in the form $$\mathbf { r } \times \mathbf { a } = \mathbf { b } ,$$ where $\mathbf { a }$ and $\mathbf { b }$ are vectors to be found.

Edexcel F3 2022 June Q8

12 marks Challenging +1.2

The plane $\Pi$ has equation

$$3 x + 4 y - z = 17$$ The line $l _ { 1 }$ is perpendicular to $\Pi$ and passes through the point $P ( - 4 , - 5,3 )$ The line $l _ { 1 }$ intersects $\Pi$ at the point $Q$

Determine the coordinates of $Q$ Given that the point $R ( - 1,6,4 )$ lies on $\Pi$
determine a Cartesian equation of the plane containing $P Q R$ The line $l _ { 2 }$ passes through $P$ and $R$ The line $l _ { 3 }$ is the reflection of $l _ { 2 }$ in $\Pi$
Determine a vector equation for $l _ { 3 }$

OCR MEI C4 Q1

17 marks Challenging +1.2

1 With respect to cartesian coordinates Oxyz, a laser beam ABC is fired from the point $\mathrm { A } ( 1,2,4 )$, and is reflected at point B off the plane with equation $x + 2 y - 3 z = 0$, as shown in Fig. 8. $\mathrm { A } ^ { \prime }$ is the point $( 2,4,1 )$, and $M$ is the midpoint of $\mathrm { AA } ^ { \prime }$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{b46db958-aa88-47fb-8db3-786472791577-1_562_716_464_650} \captionsetup{labelformat=empty} \caption{Fig. 8}

\end{figure}

Show that $\mathrm { AA } ^ { \prime }$ is perpendicular to the plane $x + 2 y - 3 z = 0$, and that M lies in the plane. The vector equation of the line AB is $\mathbf { r } = \left( \begin{array} { l } 1 \\ 2 \\ 4 \end{array} \right) + \lambda \left( \begin{array} { r } 1 \\ - 1 \\ 2 \end{array} \right)$.
Find the coordinates of B , and a vector equation of the line $\mathrm { A } ^ { \prime } \mathrm { B }$.
Given that $\mathrm { A } ^ { \prime } \mathrm { BC }$ is a straight line, find the angle $\theta$.
Find the coordinates of the point where BC crosses the Oxz plane (the plane containing the $x$ - and $z$-axes)

Edexcel CP1 2023 June Q5

12 marks Challenging +1.8

5 The line $l _ { 1 }$ has equation $\frac { x + 5 } { 1 } = \frac { y + 4 } { - 3 } = \frac { z - 3 } { 5 }$ The plane $\Pi _ { 1 }$ has equation $2 x + 3 y - 2 z = 6$

Find the point of intersection of $l _ { 1 }$ and $\Pi _ { 1 }$ The line $l _ { 2 }$ is the reflection of the line $l _ { 1 }$ in the plane $\Pi _ { 1 }$
Show that a vector equation for the line $l _ { 2 }$ is $$\mathbf { r } = \left( \begin{array} { r } - 7 \\ 2 \\ - 7 \end{array} \right) + \mu \left( \begin{array} { c } 10 \\ 6 \\ 2 \end{array} \right)$$ where $\mu$ is a scalar parameter. The plane $\Pi _ { 2 }$ contains the line $l _ { 1 }$ and the line $l _ { 2 }$
Determine a vector equation for the line of intersection of $\Pi _ { 1 }$ and $\Pi _ { 2 }$ The plane $\Pi _ { 3 }$ has equation r. $\left( \begin{array} { l } 1 \\ 1 \\ a \end{array} \right) = b$ where $a$ and $b$ are constants.
Given that the planes $\Pi _ { 1 } , \Pi _ { 2 }$ and $\Pi _ { 3 }$ form a sheaf,
determine the value of $a$ and the value of $b$.

Edexcel FP3 Q9

18 marks Standard +0.8

The plane $\Pi$ passes through the points $$A(-1, -1, 1), B(4, 2, 1) \text{ and } C(2, 1, 0).$$

Find a vector equation of the line perpendicular to $\Pi$ which passes through the point $D(1, 2, 3)$. [3]
Find the volume of the tetrahedron $ABCD$. [3]
Obtain the equation of $\Pi$ in the form $\mathbf{r} \cdot \mathbf{n} = p$. [3]

The perpendicular from $D$ to the plane $\Pi$ meets $\Pi$ at the point $E$.

Find the coordinates of $E$. [4]
Show that $DE = \frac{11\sqrt{35}}{35}$. [2]

The point $D'$ is the reflection of $D$ in $\Pi$.

Find the coordinates of $D'$. [3]

OCR FP3 2010 June Q7

12 marks Challenging +1.2

A line $l$ has equation $\mathbf{r} = \begin{pmatrix} -7 \\ -3 \\ 0 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -2 \\ 3 \end{pmatrix}$. A plane $\Pi$ passes through the points $(1, 3, 5)$ and $(5, 2, 5)$, and is parallel to $l$.

Find an equation of $\Pi$, giving your answer in the form $\mathbf{r} \cdot \mathbf{n} = p$. [4]
Find the distance between $l$ and $\Pi$. [4]
Find an equation of the line which is the reflection of $l$ in $\Pi$, giving your answer in the form $\mathbf{r} = \mathbf{a} + t\mathbf{b}$. [4]

AQA Further Paper 1 2022 June Q10

12 marks Challenging +1.8

In this question all measurements are in centimetres. A small, thin laser pen is set up with one end at $A(7, 2, -3)$ and the other end at $B(9, -3, -2)$ A laser beam travels from $A$ to $B$ and continues in a straight line towards a large thin sheet of glass. The sheet of glass lies within a plane $\Pi_1$ which is modelled by the equation $$4x + py + 5z = 9$$ where $p$ is an integer.

The laser beam hits $\Pi_1$ at an acute angle $\alpha$, where $\sin \alpha = \frac{\sqrt{15}}{75}$ Find the value of $p$ [6 marks]
A second large sheet of glass lies on the other side of $\Pi_1$ This second sheet lies within a plane $\Pi_2$ which is modelled by the equation $$4x + py + 5z = -5$$ Calculate the distance between the sheets of glass. [2 marks]
The point $A(7, 2, -3)$ is reflected in $\Pi_1$ Find the coordinates of the image of $A$ after reflection in $\Pi_1$ [4 marks]