4.04j Shortest distance: between a point and a plane

65 questions

Sort by: Default | Easiest first | Hardest first
Edexcel FP3 2014 June Q1
8 marks Standard +0.3
The line \(l\) passes through the point \(P(2, 1, 3)\) and is perpendicular to the plane \(\Pi\) whose vector equation is $$\mathbf{r} \cdot (\mathbf{i} - 2\mathbf{j} - \mathbf{k}) = 3$$ Find
  1. a vector equation of the line \(l\), [2]
  2. the position vector of the point where \(l\) meets \(\Pi\). [4]
  3. Hence find the perpendicular distance of \(P\) from \(\Pi\). [2]
Edexcel FP3 Q9
18 marks Standard +0.8
The plane \(\Pi\) passes through the points $$A(-1, -1, 1), B(4, 2, 1) \text{ and } C(2, 1, 0).$$
  1. Find a vector equation of the line perpendicular to \(\Pi\) which passes through the point \(D(1, 2, 3)\). [3]
  2. Find the volume of the tetrahedron \(ABCD\). [3]
  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 \(E\).
  1. Find the coordinates of \(E\). [4]
  2. Show that \(DE = \frac{11\sqrt{35}}{35}\). [2]
The point \(D'\) is the reflection of \(D\) in \(\Pi\).
  1. 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 \(l\).
  1. Find an equation of \(\Pi\), giving your answer in the form \(\mathbf{r} \cdot \mathbf{n} = p\). [4]
  2. Find the distance between \(l\) and \(\Pi\). [4]
  3. 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]
OCR FP3 2011 June Q1
6 marks Standard +0.3
A line \(l\) has equation \(\frac{x-1}{5} = \frac{y-6}{6} = \frac{z+3}{-7}\) and a plane \(p\) has equation \(x + 2y - z = 40\).
  1. Find the acute angle between \(l\) and \(p\). [4]
  2. Find the perpendicular distance from the point \((1, 6, -3)\) to \(p\). [2]
Edexcel AEA 2008 June Q7
22 marks Challenging +1.8
Relative to a fixed origin \(O\), the position vectors of the points \(A\), \(B\) and \(C\) are $$\overrightarrow{OA} = -3\mathbf{i} + \mathbf{j} - 9\mathbf{k}, \quad \overrightarrow{OB} = \mathbf{i} - \mathbf{k}, \quad \overrightarrow{OC} = 5\mathbf{i} + 2\mathbf{j} - 5\mathbf{k} \text{ respectively}.$$
  1. Find the cosine of angle \(ABC\). [4]
The line \(L\) is the angle bisector of angle \(ABC\).
  1. Show that an equation of \(L\) is \(\mathbf{r} = \mathbf{i} - \mathbf{k} + t(\mathbf{i} + 2\mathbf{j} - 7\mathbf{k})\). [4]
  2. Show that \(|\overrightarrow{AB}| = |\overrightarrow{AC}|\). [2]
The circle \(S\) lies inside triangle \(ABC\) and each side of the triangle is a tangent to \(S\).
  1. Find the position vector of the centre of \(S\). [7]
  2. Find the radius of \(S\). [5]
AQA Further Paper 1 2023 June Q9
9 marks Standard +0.3
The position vectors of the points \(A\), \(B\) and \(C\) are $$\mathbf{a} = 2\mathbf{i} + \mathbf{j} + 2\mathbf{k}$$ $$\mathbf{b} = -\mathbf{i} - 8\mathbf{j} + 2\mathbf{k}$$ $$\mathbf{c} = -2\mathbf{j}$$ respectively.
  1. Find the area of the triangle \(ABC\) [4 marks]
  2. The points \(A\), \(B\) and \(C\) all lie in the plane \(\Pi\) Find an equation of the plane \(\Pi\), in the form \(\mathbf{r} \cdot \mathbf{n} = d\) [2 marks]
  3. The point \(P\) has position vector \(\mathbf{p} = \mathbf{i} + 4\mathbf{j} + 2\mathbf{k}\) Find the exact distance of \(P\) from \(\Pi\) [3 marks]
Edexcel CP1 2021 June Q7
8 marks Standard +0.8
The plane \(\Pi\) has equation $$\mathbf{r} = \begin{pmatrix} 3 \\ 3 \\ 2 \end{pmatrix} + \lambda \begin{pmatrix} -1 \\ 2 \\ 1 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ 0 \\ 1 \end{pmatrix}$$ where \(\lambda\) and \(\mu\) are scalar parameters.
  1. Show that vector \(\mathbf{2i + 3j - 4k}\) is perpendicular to \(\Pi\). [2]
  2. Hence find a Cartesian equation of \(\Pi\). [2]
The line \(l\) has equation $$\mathbf{r} = \begin{pmatrix} 4 \\ -5 \\ 2 \end{pmatrix} + t \begin{pmatrix} 1 \\ 6 \\ -3 \end{pmatrix}$$ where \(t\) is a scalar parameter. The point \(A\) lies on \(l\). Given that the shortest distance between \(A\) and \(\Pi\) is \(2\sqrt{29}\)
  1. determine the possible coordinates of \(A\). [4]
SPS SPS ASFM 2020 May Q2
10 marks Standard +0.3
The position vector of point \(A\) is \(\mathbf{a} = -9\mathbf{i} + 2\mathbf{j} + 6\mathbf{k}\). The line \(l\) passes through \(A\) and is perpendicular to \(\mathbf{a}\).
  1. Determine the shortest distance between the origin, \(O\), and \(l\). [2] \(l\) is also perpendicular to the vector \(\mathbf{b}\) where \(\mathbf{b} = -2\mathbf{i} + \mathbf{j} + \mathbf{k}\).
  2. Find a vector which is perpendicular to both \(\mathbf{a}\) and \(\mathbf{b}\). [1]
  3. Write down an equation of \(l\) in vector form. [1] \(P\) is a point on \(l\) such that \(PA = 2OA\).
  4. Find angle \(POA\) giving your answer to 3 significant figures. [3] \(C\) is a point whose position vector, \(\mathbf{c}\), is given by \(\mathbf{c} = p\mathbf{a}\) for some constant \(p\). The line \(m\) passes through \(C\) and has equation \(\mathbf{r} = \mathbf{c} + \mu\mathbf{b}\). The point with position vector \(9\mathbf{i} + 8\mathbf{j} - 12\mathbf{k}\) lies on \(m\).
  5. Find the value of \(p\). [3]
SPS SPS FM Pure 2021 May Q2
8 marks Standard +0.8
  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]
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]
OCR Further Pure Core 2 2018 March Q1
8 marks Standard +0.3
Plane \(\Pi\) has equation \(3x - y + 2z = 33\). Line \(l\) has the following vector equation. $$l: \quad \mathbf{r} = \begin{pmatrix} 1 \\ 0 \\ 5 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 2 \\ 3 \end{pmatrix}$$
  1. Find the acute angle between \(\Pi\) and \(l\). [3]
  2. Find the coordinates of the point of intersection of \(\Pi\) and \(l\). [3]
  3. \(S\) is the point \((4, 5, -5)\). Find the shortest distance from \(S\) to \(\Pi\). [2]
OCR Further Pure Core 2 2018 September Q3
6 marks Standard +0.3
The equation of a plane, \(\Pi\), is $$\Pi: \mathbf{r} = \begin{pmatrix} 2 \\ -3 \\ 5 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 1 \\ 3 \end{pmatrix} + \mu \begin{pmatrix} -1 \\ 2 \\ 1 \end{pmatrix}.$$
  1. Find a vector which is perpendicular to \(\Pi\). [2]
  2. Hence find an equation for \(\Pi\) in the form \(\mathbf{r} \cdot \mathbf{n} = p\). [2]
  3. Find in the form \(\sqrt{q}\) the shortest distance between \(\Pi\) and the origin, where \(q\) is a rational number. [2]
OCR Further Additional Pure 2018 September Q4
12 marks Challenging +1.2
The points \(A\), \(B\), \(C\) and \(P\) have coordinates \((a, 0, 0)\), \((0, b, 0)\), \((0, 0, c)\) and \((a, b, c)\) respectively, where \(a\), \(b\) and \(c\) are positive constants. The plane \(\Pi\) contains \(A\), \(B\) and \(C\).
    1. Use the scalar triple product to determine
    2. Hence show that the distance from \(P\) to \(\Pi\) is twice the distance from \(O\) to \(\Pi\). [2]
    1. Determine a vector which is normal to \(\Pi\). [2]
    2. Hence determine, in terms of \(a\), \(b\) and \(c\) only, the distance from \(P\) to \(\Pi\). [3]
Pre-U Pre-U 9795/1 2011 June Q9
11 marks Standard +0.3
  1. The points \(A\), \(B\) and \(C\) have position vectors $$\mathbf{a} = \begin{pmatrix} 19 \\ 3 \\ 10 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 12 \\ 7 \\ -1 \end{pmatrix} \quad \text{and} \quad \mathbf{c} = \begin{pmatrix} 5 \\ 15 \\ 3 \end{pmatrix}$$ respectively, and \(O\) is the origin. Calculate the volume of the tetrahedron \(OABC\). [3]
    1. The plane \(\Pi_1\) has equation \(\mathbf{r} = \begin{pmatrix} 2 \\ 1 \\ 4 \end{pmatrix} + \lambda \begin{pmatrix} 3 \\ 1 \\ -1 \end{pmatrix} + \mu \begin{pmatrix} 6 \\ 2 \\ 5 \end{pmatrix}\). Determine an equation for \(\Pi_1\) in the form \(\mathbf{r} \cdot \mathbf{n} = d\). [4]
    2. A second plane, \(\Pi_2\), has equation \(\mathbf{r} \cdot \begin{pmatrix} 1 \\ 4 \\ 7 \end{pmatrix} = 13\). Find a vector equation for the line of intersection of \(\Pi_1\) and \(\Pi_2\). [4]
Pre-U Pre-U 9795 Specimen Q12
12 marks Standard +0.3
With respect to an origin \(O\), the points \(A, B, C, D\) have position vectors $$\mathbf{2i - j + k}, \quad \mathbf{i - 2k}, \quad \mathbf{-i + 3j + 2k}, \quad \mathbf{-i + j + 4k},$$ respectively. Find
  1. a vector perpendicular to the plane \(OAB\), [2]
  2. the acute angle between the planes \(OAB\) and \(OCD\), correct to the nearest \(0.1°\), [3]
  3. the shortest distance between the line which passes through \(A\) and \(B\) and the line which passes through \(C\) and \(D\), [4]
  4. the perpendicular distance from the point \(A\) to the line which passes through \(C\) and \(D\). [3]