4.04b Plane equations: cartesian and vector forms

The line \(l_1\) has equation $$\mathbf{r} = \mathbf{i} + 6\mathbf{j} - \mathbf{k} + \lambda(2\mathbf{i} + 3\mathbf{k})$$ and the line \(l_2\) has equation $$\mathbf{r} = 3\mathbf{i} + p\mathbf{j} + \mu(\mathbf{i} - 2\mathbf{j} + \mathbf{k}), \text{ where } p \text{ is a constant.}$$ The plane \(\Pi_1\) contains \(l_1\) and \(l_2\).

1. Find a vector which is normal to \(\Pi_1\). [2]
2. Show that an equation for \(\Pi_1\) is \(6x + y - 4z = 16\). [2]
3. Find the value of \(p\). [1]

The plane \(\Pi_2\) has equation \(\mathbf{r} \cdot (\mathbf{i} + 2\mathbf{j} + \mathbf{k}) = 2\).

1. Find an equation for the line of intersection of \(\Pi_1\) and \(\Pi_2\), giving your answer in the form $$(\mathbf{r} - \mathbf{a}) \times \mathbf{b} = \mathbf{0}.$$ [5]

Two lines have vector equations $$\mathbf{r} = \mathbf{i} - 2\mathbf{j} + 4\mathbf{k} + \lambda(3\mathbf{i} + \mathbf{j} + a\mathbf{k})$$ and $$\mathbf{r} = -8\mathbf{i} + 2\mathbf{j} + 3\mathbf{k} + \mu(\mathbf{i} - 2\mathbf{j} - \mathbf{k}),$$ where \(a\) is a constant.

1. Given that the lines are skew, find the value that \(a\) cannot take. [6]
2. Given instead that the lines intersect, find the point of intersection. [2]

Verify that the vector \(2\mathbf{i} - \mathbf{j} + 4\mathbf{k}\) is perpendicular to the plane through the points A(2, 0, 1), B(1, 2, 2) and C(0, -4, 1). Hence find the cartesian equation of the plane. [5]

When a light ray passes from air to glass, it is deflected through an angle. The light ray ABC starts at point A \((1, 2, 2)\), and enters a glass object at point B \((0, 0, 2)\). The surface of the glass object is a plane with normal vector \(\mathbf{n}\). Fig. 7 shows a cross-section of the glass object in the plane of the light ray and \(\mathbf{n}\). \includegraphics{figure_7}

1. Find the vector \(\overrightarrow{AB}\) and a vector equation of the line AB. [2]

The surface of the glass object is a plane with equation \(x + z = 2\). AB makes an acute angle \(\theta\) with the normal to this plane.

1. Write down the normal vector \(\mathbf{n}\), and hence calculate \(\theta\), giving your answer in degrees. [5]

The line BC has vector equation \(\mathbf{r} = \begin{pmatrix} 0 \\ 0 \\ 2 \end{pmatrix} + \mu \begin{pmatrix} -2 \\ -2 \\ -1 \end{pmatrix}\). This line makes an acute angle \(\phi\) with the normal to the plane.

1. Show that \(\phi = 45°\). [3]
2. Snell's Law states that \(\sin\theta = k\sin\phi\), where \(k\) is a constant called the refractive index. Find \(k\). [2]

The light ray leaves the glass object through a plane with equation \(x + z = -1\). Units are centimetres.

1. Find the point of intersection of the line BC with the plane \(x + z = -1\). Hence find the distance the light ray travels through the glass object. [5]

A piece of cloth ABDC is attached to the tops of vertical poles AE, BF, DG and CH, where E, F, G and H are at ground level (see Fig. 7). Coordinates are as shown, with lengths in metres. The length of pole DG is \(k\) metres. \includegraphics{figure_7}

1. Write down the vectors \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\). Hence calculate the angle BAC. [6]
2. Verify that the equation of the plane ABC is \(x + y - 2z + d = 0\), where \(d\) is a constant to be determined. Calculate the acute angle the plane makes with the horizontal plane. [7]
3. Given that A, B, D and C are coplanar, show that \(k = 3\). Hence show that ABDC is a trapezium, and find the ratio of CD to AB. [5]

With respect to cartesian coordinates \(Oxyz\), a laser beam ABC is fired from the point A(1, 2, 4), and is reflected at point B off the plane with equation \(x + 2y - 3z = 0\), as shown in Fig. 8. A' is the point (2, 4, 1), and M is the midpoint of AA'. \includegraphics{figure_8}

1. Show that AA' is perpendicular to the plane \(x + 2y - 3z = 0\), and that M lies in the plane. [4]

The vector equation of the line AB is \(\mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 4 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}\).

1. Find the coordinates of B, and a vector equation of the line A'B. [6]
2. Given that A'BC is a straight line, find the angle \(\theta\). [4]
3. Find the coordinates of the point where BC crosses the \(Oxz\) plane (the plane containing the \(x\)- and \(z\)-axes). [3]

Fig. 7 shows a tetrahedron ABCD. The coordinates of the vertices, with respect to axes Oxyz, are A(-3, 0, 0), B(2, 0, -2), C(0, 4, 0) and D(0, 4, 5). \includegraphics{figure_7}

1. Find the lengths of the edges AB and AC, and the size of the angle CAB. Hence calculate the area of triangle ABC. [7]
2. 1. Verify that 4i - 3j + 10k is normal to the plane ABC. [2]
   2. Hence find the equation of this plane. [2]
3. Write down a vector equation for the line through D perpendicular to the plane ABC. Hence find the point of intersection of this line with the plane ABC. [5]

The volume of a tetrahedron is \(\frac{1}{3} \times \text{area of base} \times \text{height}\).

1. Find the volume of the tetrahedron ABCD. [2]

The upper and lower surfaces of a coal seam are modelled as planes ABC and DEF, as shown in Fig. 8. All dimensions are metres. \includegraphics{figure_1} Relative to axes \(Ox\) (due east), \(Oy\) (due north) and \(Oz\) (vertically upwards), the coordinates of the points are as follows. A: \((0, 0, -15)\) \quad B: \((100, 0, -30)\) \quad C: \((0, 100, -25)\) D: \((0, 0, -40)\) \quad E: \((100, 0, -50)\) \quad F: \((0, 100, -35)\)

1. Verify that the cartesian equation of the plane ABC is \(3x + 2y + 20z + 300 = 0\). [3]
2. Find the vectors \(\overrightarrow{DE}\) and \(\overrightarrow{DF}\). Show that the vector \(2\mathbf{i} - \mathbf{j} + 20\mathbf{k}\) is perpendicular to each of these vectors. Hence find the cartesian equation of the plane DEF. [6]
3. By calculating the angle between their normal vectors, find the angle between the planes ABC and DEF. [4]

It is decided to drill down to the seam from a point R \((15, 34, 0)\) in a line perpendicular to the upper surface of the seam. This line meets the plane ABC at the point S.

1. Write down a vector equation of the line RS. Find the coordinates of S. [5]

Write down normal vectors to the planes \(2x + 3y + 4z = 10\) and \(x - 2y + z = 5\). Hence show that these planes are perpendicular to each other. [4]

A computer-controlled machine can be programmed to make cuts by entering the equation of the plane of the cut, and to drill holes by entering the equation of the line of the hole. A \(20\text{ cm} \times 30\text{ cm} \times 30\text{ cm}\) cuboid is to be cut and drilled. The cuboid is positioned relative to \(x\)-, \(y\)- and \(z\)-axes as shown in Fig. 8.1. \includegraphics{figure_2} First, a plane cut is made to remove the corner at E. The cut goes through the points P, Q and R, which are the midpoints of the sides ED, EA and EF respectively.

1. Write down the coordinates of P, Q and R. Hence show that \(\overrightarrow{PQ} = \begin{pmatrix} 0 \\ 0 \\ -15 \end{pmatrix}\) and \(\overrightarrow{PR} = \begin{pmatrix} -15 \\ 0 \\ 1 \end{pmatrix}\). [4]
2. Show that \(\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}\) is perpendicular to the plane through P, Q and R. Hence find the cartesian equation of this plane. [5]

A hole is then drilled perpendicular to triangle PQR, as shown in Fig. 8.2. The hole passes through the triangle at the point T which divides the line PS in the ratio \(2:1\), where S is the midpoint of QR.

1. Write down the coordinates of S, and show that the point T has coordinates \((-5, 16, 25)\). [4]
2. Write down a vector equation of the line of the drill hole. Hence determine whether or not this line passes through C. [5]

A tent has vertices ABCDEF with coordinates as shown in Fig. 7. Lengths are in metres. The \(Oxy\) plane is horizontal. \includegraphics{figure_3}

1. Find the length of the ridge of the tent DE, and the angle this makes with the horizontal. [4]
2. Show that the vector \(\mathbf{i} - 4\mathbf{j} + 5\mathbf{k}\) is normal to the plane through A, D and E. Hence find the equation of this plane. Given that B lies in this plane, find \(a\). [7]
3. Verify that the equation of the plane BCD is \(x + z = 8\). Hence find the acute angle between the planes ABDE and BCD. [6]

Lines \(l_1\) and \(l_2\) have equations $$\frac{x-3}{2} = \frac{y-4}{-1} = \frac{z+1}{1} \quad \text{and} \quad \frac{x-5}{4} = \frac{y-1}{3} = \frac{z-1}{2}$$ respectively.

1. Find the equation of the plane \(\Pi_1\) which contains \(l_1\) and is parallel to \(l_2\), giving your answer in the form \(\mathbf{r} \cdot \mathbf{n} = p\). [5]
2. Find the equation of the plane \(\Pi_2\) which contains \(l_2\) and is parallel to \(l_1\), giving your answer in the form \(\mathbf{r} \cdot \mathbf{n} = p\). [2]
3. Find the distance between the planes \(\Pi_1\) and \(\Pi_2\). [2]
4. State the relationship between the answer to part (iii) and the lines \(l_1\) and \(l_2\). [1]

A tetrahedron \(ABCD\) is such that \(AB\) is perpendicular to the base \(BCD\). The coordinates of the points \(A, C\) and \(D\) are \((-1, -7, 2), (5, 0, 3)\) and \((-1, 3, 3)\) respectively, and the equation of the plane \(BCD\) is \(x + 2y - 2z = -1\).

1. Find, in either order, the coordinates of \(B\) and the length of \(AB\). [5]
2. Find the acute angle between the planes \(ACD\) and \(BCD\). [6]

Two lines have equations $$\frac{x-k}{2} = \frac{y+1}{-5} = \frac{z-1}{-3} \quad \text{and} \quad \frac{x-k}{1} = \frac{y+4}{-4} = \frac{z}{-2},$$ where \(k\) is a constant.

1. Show that, for all values of \(k\), the lines intersect, and find their point of intersection in terms of \(k\). [6]
2. For the case \(k = 1\), find the equation of the plane in which the lines lie, giving your answer in the form \(ax + by + cz = d\). [4]

\includegraphics{figure_6} The cuboid \(OABCDEFG\) shown in the diagram has \(\overrightarrow{OA} = 4\mathbf{i}, \overrightarrow{OC} = 2\mathbf{j}, \overrightarrow{OD} = 3\mathbf{k}\), and \(M\) is the mid-point of \(GF\).

1. Find the equation of the plane \(ACGE\), giving your answer in the form \(\mathbf{r} \cdot \mathbf{n} = p\). [4]
2. The plane \(OEFC\) has equation \(\mathbf{r} \cdot (3\mathbf{i} - 4\mathbf{k}) = 0\). Find the acute angle between the planes \(OEFC\) and \(ACGE\). [4]
3. The line \(AM\) meets the plane \(OEFC\) at the point \(W\). Find the ratio \(AW : WM\). [5]

A line \(l\) has equation \(\frac{x - 6}{-4} = \frac{y + 7}{8} = \frac{z + 10}{7}\) and a plane \(p\) has equation \(3x - 4y - 2z = 8\).

1. Find the point of intersection of \(l\) and \(p\). [3]
2. Find the equation of the plane which contains \(l\) and is perpendicular to \(p\), giving your answer in the form \(ax + by + cz = d\). [5]

The plane \(\Pi_1\) has equation \(\mathbf{r} = \begin{pmatrix} 2 \\ 2 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} + \mu \begin{pmatrix} -5 \\ -2 \end{pmatrix}\).

1. Express the equation of \(\Pi_1\) in the form \(\mathbf{r} \cdot \mathbf{n} = p\). [4] The plane \(\Pi_2\) has equation \(\mathbf{r} \cdot \begin{pmatrix} 7 \\ 1 \\ -3 \end{pmatrix} = 21\).
2. Find an equation of the line of intersection of \(\Pi_1\) and \(\Pi_2\), giving your answer in the form \(\mathbf{r} = \mathbf{a} + t\mathbf{b}\). [5]

A tetrahedron \(ABCD\) is such that \(AB\) is perpendicular to the base \(BCD\). The coordinates of the points \(A\), \(C\) and \(D\) are \((-1, -7, 2)\), \((5, 0, 3)\) and \((-1, 3, 3)\) respectively, and the equation of the plane \(BCD\) is \(x + 2y - 2z = -1\).

1. Find, in either order, the coordinates of \(B\) and the length of \(AB\). [5]
2. Find the acute angle between the planes \(ACD\) and \(BCD\). [6]

A regular tetrahedron has vertices at the points $$A\left(0, 0, \frac{2}{\sqrt{3}}\sqrt{6}\right), \quad B\left(\frac{2}{\sqrt{3}}\sqrt{3}, 0, 0\right), \quad C\left(-\frac{1}{3}\sqrt{3}, 1, 0\right), \quad D\left(-\frac{1}{3}\sqrt{3}, -1, 0\right).$$

1. Obtain the equation of the face \(ABC\) in the form $$x + \sqrt{3}y + \left(\frac{1}{2}\sqrt{2}\right)z = \frac{2}{3}\sqrt{3}.$$ [5] (Answers which only verify the given equation will not receive full credit.)
2. Give a geometrical reason why the equation of the face \(ABD\) can be expressed as $$x - \sqrt{3}y + \left(\frac{1}{2}\sqrt{2}\right)z = \frac{2}{3}\sqrt{3}.$$ [2]
3. Hence find the cosine of the angle between two faces of the tetrahedron. [4]

Two intersecting lines, lying in a plane \(p\), have equations $$\frac{x-1}{2} = \frac{y-3}{1} = \frac{z-4}{-3} \quad \text{and} \quad \frac{x-1}{-1} = \frac{y-3}{2} = \frac{z-4}{4}.$$

1. Obtain the equation of \(p\) in the form \(2x - y + z = 3\). [3]
2. Plane \(q\) has equation \(2x - y + z = 21\). Find the distance between \(p\) and \(q\). [3]

Three planes \(\Pi_1\), \(\Pi_2\) and \(\Pi_3\) have equations $$\mathbf{r} \cdot (\mathbf{i} + \mathbf{j} - 2\mathbf{k}) = 5, \quad \mathbf{r} \cdot (\mathbf{i} - \mathbf{j} + 3\mathbf{k}) = 6, \quad \mathbf{r} \cdot (\mathbf{i} + 5\mathbf{j} - 12\mathbf{k}) = 12,$$ respectively. Planes \(\Pi_1\) and \(\Pi_2\) intersect in a line \(l\); planes \(\Pi_2\) and \(\Pi_3\) intersect in a line \(m\).

1. Show that \(l\) and \(m\) are in the same direction. [5]
2. Write down what you can deduce about the line of intersection of planes \(\Pi_1\) and \(\Pi_3\). [1]
3. By considering the cartesian equations of \(\Pi_1\), \(\Pi_2\) and \(\Pi_3\), or otherwise, determine whether or not the three planes have a common line of intersection. [4]

A line \(l_1\) has equation \(\frac{x}{2} = \frac{y + 4}{3} = \frac{z + 9}{5}\).

1. Find the cartesian equation of the plane which is parallel to \(l_1\) and which contains the points \((2, 1, 5)\) and \((0, -1, 5)\). [5]
2. Write down the position vector of a point on \(l_1\) with parameter \(t\). [1]
3. Hence, or otherwise, find an equation of the line \(l_2\) which intersects \(l_1\) at right angles and which passes through the point \((-5, 3, 4)\). Give your answer in the form \(\frac{x - a}{p} = \frac{y - b}{q} = \frac{z - c}{r}\). [4]

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]

A plane has equation \(\mathbf{r} \cdot \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} = 7\) A line has equation \(\mathbf{r} = \begin{pmatrix} 2 \\ 0 \\ 1 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}\) Calculate the acute angle between the line and the plane. Give your answer to the nearest \(0.1°\) [3 marks]