Perpendicular distance from point to plane

A question is this type if and only if it asks to find the shortest distance from a point to a plane using the formula |ax₀+by₀+cz₀-d|/√(a²+b²+c²).

6 questions · Standard +0.6

4.04j Shortest distance: between a point and a plane
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Edexcel F3 2021 January Q7
11 marks Standard +0.8
  1. The point \(P\) has coordinates \(( 1,2,1 )\)
The line \(l\) has Cartesian equation $$\frac { x - 3 } { 5 } = \frac { y + 1 } { 3 } = \frac { z + 5 } { - 8 }$$ The plane \(\Pi _ { 1 }\) contains the point \(P\) and the line \(l\).
  1. Show that a Cartesian equation for \(\Pi _ { 1 }\) is $$6 x - 2 y + 3 z = 5$$ The point \(Q\) has coordinates \(( 2 , k , - 7 )\), where \(k\) is a constant.
  2. Show that the shortest distance between \(\Pi _ { 1 }\) and \(Q\) is $$\frac { 2 } { 7 } | k + 7 |$$ The plane \(\Pi _ { 2 }\) has Cartesian equation \(8 x - 4 y + z = - 3\) Given that the shortest distance between \(\Pi _ { 1 }\) and \(Q\) is the same as the shortest distance between \(\Pi _ { 2 }\) and \(Q\),
  3. determine the possible values of \(k\).
Edexcel FP3 2013 June Q8
14 marks Standard +0.3
  1. The plane \(\Pi _ { 1 }\) has vector equation
$$\mathbf { r } . ( 3 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k } ) = 5$$
  1. Find the perpendicular distance from the point \(( 6,2,12 )\) to the plane \(\Pi _ { 1 }\) The plane \(\Pi _ { 2 }\) has vector equation $$\mathbf { r } = \lambda ( 2 \mathbf { i } + \mathbf { j } + 5 \mathbf { k } ) + \mu ( \mathbf { i } - \mathbf { j } - 2 \mathbf { k } ) , \text { where } \lambda \text { and } \mu \text { are scalar parameters. }$$
  2. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) giving your answer to the nearest degree.
  3. Find an equation of the line of intersection of the two planes in the form \(\mathbf { r } \times \mathbf { a } = \mathbf { b }\), where \(\mathbf { a }\) and \(\mathbf { b }\) are constant vectors.
CAIE FP1 2017 Specimen Q11 EITHER
Challenging +1.8
The points \(A , B\) and \(C\) have position vectors \(\mathbf { i } , 2 \mathbf { j }\) and \(4 \mathbf { k }\) respectively, relative to an origin \(O\). The point \(N\) is the foot of the perpendicular from \(O\) to the plane \(A B C\). The point \(P\) on the line-segment \(O N\) is such that \(O P = \frac { 3 } { 4 } O N\). The line \(A P\) meets the plane \(O B C\) at \(Q\).
  1. Find a vector perpendicular to the plane \(A B C\) and show that the length of \(O N\) is \(\frac { 4 } { \sqrt { } ( 21 ) }\).
  2. Find the position vector of the point \(Q\).
  3. Show that the acute angle between the planes \(A B C\) and \(A B Q\) is \(\cos ^ { - 1 } \left( \frac { 2 } { 3 } \right)\).
OCR MEI Further Pure Core 2024 June Q11
14 marks Standard +0.3
11 The plane \(\Pi\) has equation \(2 x - y + 2 z = 4\). The point \(P\) has coordinates \(( 8,4,5 )\).
  1. Calculate the shortest distance from P to \(\Pi\). The line \(L\) has equation \(\frac { x - 2 } { 3 } = \frac { y } { 2 } = \frac { z + 3 } { 4 }\).
  2. Verify that P lies on L .
  3. Find the coordinates of the point of intersection of L and \(\Pi\).
  4. Determine the acute angle between L and \(\Pi\).
  5. Use the results of parts (b), (c) and (d) to verify your answer to part (a).
Edexcel CP2 Specimen Q2
8 marks Standard +0.3
  1. The plane \(\Pi _ { 1 }\) has vector equation
$$\mathbf { r } \cdot ( 3 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k } ) = 5$$
  1. Find the perpendicular distance from the point \(( 6,2,12 )\) to the plane \(\Pi _ { 1 }\) The plane \(\Pi _ { 2 }\) has vector equation $$\mathbf { r } = \lambda ( 2 \mathbf { i } + \mathbf { j } + 5 \mathbf { k } ) + \mu ( \mathbf { i } - \mathbf { j } - 2 \mathbf { k } )$$ where \(\lambda\) and \(\mu\) are scalar parameters.
  2. Show that the vector \(- \mathbf { i } - 3 \mathbf { j } + \mathbf { k }\) is perpendicular to \(\Pi _ { 2 }\)
  3. Show that the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) is \(52 ^ { \circ }\) to the nearest degree.
OCR Further Pure Core 2 2021 June Q1
8 marks Standard +0.3
  1. A plane \(\Pi\) has the equation \(\mathbf{r} \cdot \begin{pmatrix} 3 \\ 6 \\ -2 \end{pmatrix} = 15\). \(C\) is the point \((4, -5, 1)\). Find the shortest distance between \(\Pi\) and \(C\). [3]
  2. Lines \(l_1\) and \(l_2\) have the following equations. \(l_1: \mathbf{r} = \begin{pmatrix} 4 \\ 3 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} -2 \\ 4 \\ -2 \end{pmatrix}\) \(l_2: \mathbf{r} = \begin{pmatrix} 5 \\ 2 \\ 4 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix}\) Find, in exact form, the distance between \(l_1\) and \(l_2\). [5]