Distance between parallel planes or line and parallel plane

CAIE P3 2017 June Q6

8 marks Standard +0.3

6 The plane with equation $2 x + 2 y - z = 5$ is denoted by $m$. Relative to the origin $O$, the points $A$ and $B$ have coordinates $( 3,4,0 )$ and $( - 1,0,2 )$ respectively.

Show that the plane $m$ bisects $A B$ at right angles.
A second plane $p$ is parallel to $m$ and nearer to $O$. The perpendicular distance between the planes is 1 .
Find the equation of $p$, giving your answer in the form $a x + b y + c z = d$.

Edexcel F3 2015 June Q7

11 marks Standard +0.8

The plane $\Pi _ { 1 }$ contains the point $( 3,3 , - 2 )$ and the line $\frac { x - 1 } { 2 } = \frac { y - 2 } { - 1 } = \frac { z + 1 } { 4 }$
1. Show that a cartesian equation of the plane $\Pi _ { 1 }$ is
$$3 x - 10 y - 4 z = - 13$$ The plane $\Pi _ { 2 }$ is parallel to the plane $\Pi _ { 1 }$ The point ( $\alpha , 1,1$ ), where $\alpha$ is a constant, lies in $\Pi _ { 2 }$ Given that the shortest distance between the planes $\Pi _ { 1 }$ and $\Pi _ { 2 }$ is $\frac { 1 } { \sqrt { 5 } }$
find the possible values of $\alpha$.

OCR FP3 2014 June Q6

8 marks Standard +0.8

6 The line $l$ has equations $\frac { x - 1 } { 2 } = \frac { y + 2 } { 3 } = \frac { z - 7 } { 5 }$. The plane $\Pi$ has equation $4 x - y - z = 8$.

Show that $l$ is parallel to $\Pi$ but does not lie in $\Pi$.
The point $A ( 1 , - 2,7 )$ is on $l$. Write down a vector equation of the line through $A$ which is perpendicular to $\Pi$. Hence find the position vector of the point on $\Pi$ which is closest to $A$.
Hence write down a vector equation of the line in $\Pi$ which is parallel to $l$ and closest to it.

OCR FP3 2010 June Q7

12 marks Challenging +1.2

7 A line $l$ has equation $\mathbf { r } = \left( \begin{array} { r } - 7 \\ - 3 \\ 0 \end{array} \right) + \lambda \left( \begin{array} { r } 2 \\ - 2 \\ 3 \end{array} \right)$. 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 r.n $= p$.
Find the distance between $l$ and $\Pi$.
Find an equation of the line which is the reflection of $l$ in $\Pi$, giving your answer in the form $\mathbf { r } = \mathbf { a } + t \mathbf { b }$.

AQA Further Paper 1 2020 June Q11

11 marks Standard +0.8

11 The lines $l _ { 1 } , l _ { 2 }$ and $l _ { 3 }$ are defined as follows. $$\begin{aligned} & l _ { 1 } : \left( \mathbf { r } - \left[ \begin{array} { c } 1 \\ 5 \\ - 1 \end{array} \right] \right) \times \left[ \begin{array} { c } - 2 \\ 1 \\ - 3 \end{array} \right] = \mathbf { 0 } \\ & l _ { 2 } : \left( \mathbf { r } - \left[ \begin{array} { c } - 3 \\ 2 \\ 7 \end{array} \right] \right) \times \left[ \begin{array} { c } 2 \\ - 1 \\ 3 \end{array} \right] = \mathbf { 0 } \\ & l _ { 3 } : \left( \mathbf { r } - \left[ \begin{array} { c } - 5 \\ 12 \\ - 4 \end{array} \right] \right) \times \left[ \begin{array} { l } 4 \\ 0 \\ 9 \end{array} \right] = \mathbf { 0 } \end{aligned}$$ 11

Explain how you know that two of the lines are parallel.

11

(ii)

Show that the perpendicular distance between these two parallel lines is 7.95 units, correct to three significant figures.

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11

Show that the lines $l _ { 1 }$ and $l _ { 3 }$ meet, and find the coordinates of their point of intersection. \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-23_2488_1716_219_153}