4.04b Plane equations: cartesian and vector forms

The line \(L_1\) has equation \(\mathbf{r} = \begin{pmatrix} 2 \\ 2 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix}\) The line \(L_2\) has equation \(\mathbf{r} = \begin{pmatrix} 6 \\ 4 \\ 1 \end{pmatrix} + \mu \begin{pmatrix} -2 \\ 1 \\ 1 \end{pmatrix}\)

1. Find the acute angle between the lines \(L_1\) and \(L_2\), giving your answer to the nearest 0.1° [3 marks]
2. The lines \(L_1\) and \(L_2\) lie in the plane \(\Pi_1\)
   1. Find the equation of \(\Pi_1\), giving your answer in the form \(\mathbf{r} \cdot \mathbf{n} = d\) [4 marks]
   2. Hence find the shortest distance of the plane \(\Pi_1\) from the origin. [1 mark]
3. The points \(A(4, -1, -1)\), \(B(1, 5, -7)\) and \(C(3, 4, -8)\) lie in the plane \(\Pi_2\) Find the angle between the planes \(\Pi_1\) and \(\Pi_2\), giving your answer to the nearest 0.1° [4 marks]

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.

1. 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]
2. 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]
3. 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]

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]

The points \(A\), \(B\) and \(C\) have coordinates \(A(5, 3, 4)\), \(B(8, -1, 9)\) and \(C(12, 5, 10)\) The points \(A\), \(B\) and \(C\) lie in the plane \(\Pi\)

1. Find a vector that is normal to the plane \(\Pi\) [3 marks]
2. Find a Cartesian equation of the plane \(\Pi\) [2 marks]

A line has Cartesian equations \(x - p = \frac{y + 2}{q} = 3 - z\) and a plane has equation \(\mathbf{r} \cdot \begin{bmatrix} 1 \\ -1 \\ -2 \end{bmatrix} = -3\)

1. In the case where the plane fully contains the line, find the values of \(p\) and \(q\). [3 marks]
2. In the case where the line intersects the plane at a single point, find the range of values of \(p\) and \(q\). [3 marks]
3. In the case where the angle \(\theta\) between the line and the plane satisfies \(\sin \theta = \frac{1}{\sqrt{6}}\) and the line intersects the plane at \(z = 0\)
   1. Find the value of \(q\). [4 marks]
   2. Find the value of \(p\). [3 marks]

The points \(A\), \(B\) and \(C\) have coordinates \(A(4, 5, 2)\), \(B(-3, 2, -4)\) and \(C(2, 6, 1)\)

1. Use a vector product to show that the area of triangle \(ABC\) is \(\frac{5\sqrt{11}}{2}\) [4 marks]
2. The points \(A\), \(B\) and \(C\) lie in a plane. Find a vector equation of the plane in the form \(\mathbf{r} \cdot \mathbf{n} = k\) [1 mark]
3. Hence find the exact distance of the plane from the origin. [1 mark]

The points \(A(7, 2, 8)\), \(B(7, -4, 0)\) and \(C(3, 3.2, 9.6)\) all lie in the plane \(\Pi\).

1. Find a Cartesian equation of the plane \(\Pi\). [3 marks]
2. The line \(L_1\) has equation \(\mathbf{r} = \begin{bmatrix} 5 \\ -0.4 \\ 4.8 \end{bmatrix} + \mu \begin{bmatrix} 15 \\ 3 \\ 4 \end{bmatrix}\)
   1. Show that \(L_1\) lies in the plane \(\Pi\). [2 marks]
   2. Show that every point on \(L_1\) is equidistant from \(B\) and \(C\). [4 marks]
3. The line \(L_2\) lies in the plane \(\Pi\), and every point on \(L_2\) is equidistant from \(A\) and \(B\). Find an equation of the line \(L_2\) [4 marks]
4. The points \(A\), \(B\) and \(C\) all lie on a circle \(G\). The point \(D\) is the centre of circle \(G\). Find the coordinates of \(D\). [3 marks]

A designer is using a computer aided design system to design part of a building. He models part of a roof as a triangular prism \(ABCDEF\) with parallel triangular ends \(ABC\) and \(DEF\), and a rectangular base \(ACFD\). He uses the metre as the unit of length. \includegraphics{figure_16} The coordinates of \(B\), \(C\) and \(D\) are \((3, 1, 11)\), \((9, 3, 4)\) and \((-4, 12, 4)\) respectively. He uses the equation \(x - 3y = 0\) for the plane \(ABC\). He uses \(\mathbf{r} - \begin{pmatrix} -4 \\ 12 \\ 4 \end{pmatrix} \times \begin{pmatrix} 4 \\ -12 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}\) for the equation of the line \(AD\). Find the volume of the space enclosed inside this section of the roof. [9 marks]

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]

Three planes have equations \begin{align} -x + 2y + z &= 0
2x - y - z &= 0
x + y &= a \end{align} where \(a\) is a constant.

1. Investigate the arrangement of the planes:
   [6]
2. Chris claims that the position vectors \(-\mathbf{i} + 2\mathbf{j} + \mathbf{k}\), \(2\mathbf{i} - \mathbf{j} - \mathbf{k}\) and \(\mathbf{i} + \mathbf{j}\) lie in a plane. Determine whether or not Chris is correct. [2]

The plane \(\Pi\) has equation \(3x - 5y + z = 9\).

1. Show that \(\Pi\) contains
   and
   [4]
2. Determine the equation of a plane which is perpendicular to \(\Pi\) and which passes through \((4,1,2)\). [3]

Find the cartesian equation of the plane which contains the three points \((1, 0, -1)\), \((2, 2, 1)\) and \((1, 1, 2)\). [5]

A surface \(S\) has equation \(g(x, y, z) = 0\), where \(g(x, y, z) = (y - 2x)(y + z)^2 - 18\).

1. Show that \(\frac{\partial g}{\partial y} = (y + z)(-4x + 3y + z)\). [2]
2. Show that \(\frac{\partial g}{\partial x} + 2\frac{\partial g}{\partial y} - 2\frac{\partial g}{\partial z} = 0\). [4]
3. Hence identify a vector which lies in the tangent plane of every point on \(S\), explaining your reasoning. [3]
4. Find the cartesian equation of the tangent plane to the surface \(S\) at the point P\((1, 4, -7)\). [3]

The tangent plane to the surface \(S\) at the point Q\((0, 2, 1)\) has equation \(6x - 7y - 4z = -18\).

1. Find a vector equation for the line of intersection of the tangent planes at P and Q. [4]

The line \(L\) passes through the points A\((1, 2, 3)\) and B\((2, 3, 1)\).

1. 1. Find the vector \(\overrightarrow{AB}\).
   2. Write down the vector equation of the line \(L\). [3]
2. The plane \(\Pi\) has equation \(x + 3y - 2z = 5\).
   1. Find the coordinates of the point of intersection of \(L\) and \(\Pi\).
   2. Find the acute angle between \(L\) and \(\Pi\). [9]

Points \(A\), \(B\) and \(C\) have coordinates \((0, 1, -4)\), \((1, 1, -2)\) and \((3, 2, 5)\) respectively.

1. Find the vector product \(\overrightarrow{AB} \times \overrightarrow{AC}\). [3]
2. Hence find the equation of the plane \(ABC\) in the form \(ax + by + cz = d\). [2]

The line \(l_1\) has equation \(\mathbf{r} = \begin{pmatrix} 1 \\ -3 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 3 \\ 2 \\ -2 \end{pmatrix}\). The plane \(\Pi\) has equation \(\mathbf{r} \cdot \begin{pmatrix} 2 \\ 1 \\ -3 \end{pmatrix} = 4\).

1. Find the position vector of the point of intersection of \(l_1\) and \(\Pi\). [3]
2. Find the acute angle between \(l_1\) and \(\Pi\). [3]

\(A\) is the point on \(l_1\) where \(\lambda = 1\). \(l_2\) is the line with the following properties. • \(l_2\) passes through \(A\) • \(l_2\) is perpendicular to \(l_1\) • \(l_2\) is parallel to \(\Pi\)

1. Find, in vector form, the equation of \(l_2\). [3]

Points \(A\), \(B\) and \(C\) have coordinates \((4, 2, 0)\), \((1, 5, 3)\) and \((1, 4, -2)\) respectively. The line \(l\) passes through \(A\) and \(B\).

1. Find a cartesian equation for \(l\). [3]

\(M\) is the point on \(l\) that is closest to \(C\).

1. Find the coordinates of \(M\). [4]
2. Find the exact area of the triangle \(ABC\). [4]

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 \(2\mathbf{i} + 3\mathbf{j} - 4\mathbf{k}\) 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]

The plane \(x + 2y + cz = 4\) is perpendicular to the plane \(2x - cy + 6z = 9\), where \(c\) is a constant. Find the value of \(c\). [3]

The plane \(\Pi_1\) has equation $$\mathbf{r} = 2\mathbf{i} + 4\mathbf{j} - \mathbf{k} + \lambda (\mathbf{i} + 2\mathbf{j} - 3\mathbf{k}) + \mu(-\mathbf{i} + 2\mathbf{j} + \mathbf{k})$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Find a Cartesian equation for \(\Pi_1\) [4]

The line \(l\) has equation $$\frac{x-1}{5} = \frac{y-3}{-3} = \frac{z+2}{4}$$

1. Find the coordinates of the point of intersection of \(l\) with \(\Pi_1\) [3]

The plane \(\Pi_2\) has equation $$\mathbf{r}.(2\mathbf{i} - \mathbf{j} + 3\mathbf{k}) = 5$$

1. Find, to the nearest degree, the acute angle between \(\Pi_1\) and \(\Pi_2\) [2]

The equations of two intersecting lines \(l_1\) and \(l_2\) are $$l_1: \mathbf{r} = \begin{pmatrix} 1 \\ 0 \\ a \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 1 \\ -3 \end{pmatrix}$$ $$l_2: \mathbf{r} = \begin{pmatrix} 7 \\ 9 \\ -2 \end{pmatrix} + \mu \begin{pmatrix} -1 \\ 1 \\ 2 \end{pmatrix}$$ where \(a\) is a constant. The equation of the plane \(\Pi\) is $$\mathbf{r} \cdot \begin{pmatrix} 1 \\ 5 \\ 3 \end{pmatrix} = -14.$$ \(l_1\) and \(\Pi\) intersect at \(Q\). \(l_2\) and \(\Pi\) intersect at \(R\).

1. Verify that the coordinates of \(R\) are \((13, 3, -14)\). [2]
2. Determine the exact value of the length of \(QR\). [7]

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]

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]