4.04a Line equations: 2D and 3D, cartesian and vector forms

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]

Relative to a fixed origin, two lines have the equations $$\mathbf{r} = \begin{pmatrix} 4 \\ 1 \\ 1 \end{pmatrix} + s \begin{pmatrix} 1 \\ 4 \\ 5 \end{pmatrix}$$ and $$\mathbf{r} = \begin{pmatrix} -3 \\ 1 \\ -6 \end{pmatrix} + t \begin{pmatrix} 3 \\ a \\ b \end{pmatrix},$$ where \(a\) and \(b\) are constants and \(s\) and \(t\) are scalar parameters. Given that the two lines are perpendicular,

1. find a linear relationship between \(a\) and \(b\). [2]

Given also that the two lines intersect,

1. find the values of \(a\) and \(b\), [8]
2. find the coordinates of the point where they intersect. [2]

Relative to a fixed origin, two lines have the equations $$\mathbf{r} = (7\mathbf{i} - 4\mathbf{k}) + s(4\mathbf{i} - 3\mathbf{j} + \mathbf{k}),$$ and $$\mathbf{r} = (-7\mathbf{i} + \mathbf{j} + 8\mathbf{k}) + t(-3\mathbf{i} + 2\mathbf{k}),$$ where \(s\) and \(t\) are scalar parameters.

1. Show that the two lines intersect and find the position vector of the point where they meet. [5]
2. Find, in degrees to 1 decimal place, the acute angle between the lines. [4]

Relative to a fixed origin, the points \(A\) and \(B\) have position vectors \(\begin{pmatrix} -4 \\ 1 \\ 3 \end{pmatrix}\) and \(\begin{pmatrix} -3 \\ 6 \\ 1 \end{pmatrix}\) respectively.

1. Find a vector equation for the line \(l_1\) which passes through \(A\) and \(B\). [2]

The line \(l_2\) has vector equation $$\mathbf{r} = \begin{pmatrix} 3 \\ -7 \\ 9 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ -3 \\ 1 \end{pmatrix}.$$

1. Show that lines \(l_1\) and \(l_2\) do not intersect. [5]
2. Find the position vector of the point \(C\) on \(l_2\) such that \(\angle ABC = 90°\). [6]

The line \(l_1\) passes through the points \(A\) and \(B\) with position vectors \((\mathbf{3i} + \mathbf{6j} - \mathbf{8k})\) and \((\mathbf{8j} - \mathbf{6k})\) respectively, relative to a fixed origin.

1. Find a vector equation for \(l_1\). [2]

The line \(l_2\) has vector equation $$\mathbf{r} = (-\mathbf{2i} + \mathbf{10j} + \mathbf{6k}) + \mu(\mathbf{7i} - \mathbf{4j} + \mathbf{6k}),$$ where \(\mu\) is a scalar parameter.

1. Show that lines \(l_1\) and \(l_2\) intersect. [4]
2. Find the coordinates of the point where \(l_1\) and \(l_2\) intersect. [2]

The point \(C\) lies on \(l_2\) and is such that \(AC\) is perpendicular to \(AB\).

1. Find the position vector of \(C\). [6]

Relative to a fixed origin, two lines have the equations $$\mathbf{r} = (7\mathbf{i} - 4\mathbf{k}) + s(4\mathbf{i} - 3\mathbf{j} + \mathbf{k}),$$ and $$\mathbf{r} = (-7\mathbf{i} + \mathbf{j} + 8\mathbf{k}) + t(-3\mathbf{i} + 2\mathbf{k}),$$ where \(s\) and \(t\) are scalar parameters.

1. Show that the two lines intersect and find the position vector of the point where they meet. [5]
2. Find, in degrees to 1 decimal place, the acute angle between the lines. [4]

Relative to a fixed origin, \(O\), the line \(l\) has the equation $$\mathbf{r} = \begin{pmatrix} 1 \\ p \\ -5 \end{pmatrix} + \lambda \begin{pmatrix} 3 \\ -1 \\ q \end{pmatrix},$$ where \(p\) and \(q\) are constants and \(\lambda\) is a scalar parameter. Given that the point \(A\) with coordinates \((-5, 9, -9)\) lies on \(l\),

1. find the values of \(p\) and \(q\), [3]
2. show that the point \(B\) with coordinates \((25, -1, 11)\) also lies on \(l\). [2]

The point \(C\) lies on \(l\) and is such that \(OC\) is perpendicular to \(l\).

1. Find the coordinates of \(C\). [3]
2. Find the ratio \(AC : CB\) [2]

A straight road passes through villages at the points \(A\) and \(B\) with position vectors \((9\mathbf{i} - 8\mathbf{j} + 2\mathbf{k})\) and \((4\mathbf{j} + \mathbf{k})\) respectively, relative to a fixed origin. The road ends at a junction at the point \(C\) with another straight road which lies along the line with equation $$\mathbf{r} = (2\mathbf{i} + 16\mathbf{j} - \mathbf{k}) + t(-5\mathbf{i} + 3\mathbf{j}),$$ where \(t\) is a scalar parameter.

1. Find the position vector of \(C\). [5]

Given that 1 unit on each coordinate axis represents 200 metres,

1. find the distance, in kilometres, from the village at \(A\) to the junction at \(C\). [4]

Verify that the point \((-1, 6, 5)\) lies on both the lines $$\mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \quad \text{and} \quad \mathbf{r} = \begin{pmatrix} 0 \\ 6 \\ 3 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}.$$ Find the acute angle between the lines. [7]

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]

A particle \(A\) has constant velocity \((3\mathbf{i} + \mathbf{j})\) m s\(^{-1}\) and a particle \(B\) has constant velocity \((\mathbf{i} - \mathbf{k})\) m s\(^{-1}\). At time \(t = 0\) seconds, the position vectors of the particles \(A\) and \(B\) with respect to a fixed origin \(O\) are \((-6\mathbf{i} + 4\mathbf{j} - 3\mathbf{k})\) m and \((-2\mathbf{i} + 2\mathbf{j} + 3\mathbf{k})\) m respectively.

1. Show that, in the subsequent motion, the minimum distance between \(A\) and \(B\) is \(4\sqrt{2}\) m. [6]
2. Find the position vector of \(A\) at the instant when the distance between \(A\) and \(B\) is a minimum. [2]

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]

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]

Determine whether the lines $$\frac{x-1}{-1} = \frac{y+2}{2} = \frac{z+4}{2} \quad \text{and} \quad \frac{x+3}{2} = \frac{y-1}{3} = \frac{z-5}{4}$$ intersect or are skew. [5]

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]

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 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]

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]

Line \(l_1\) has Cartesian equation $$x - 3 = \frac{2y + 2}{3} = 2 - z$$

1. Write the equation of line \(l_1\) in the form $$\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}$$ where \(\lambda\) is a parameter and \(\mathbf{a}\) and \(\mathbf{b}\) are vectors to be found. [2 marks]
2. Line \(l_2\) passes through the points \(P(3, 2, 0)\) and \(Q(n, 5, n)\), where \(n\) is a constant.
   1. Show that the lines \(l_1\) and \(l_2\) are not perpendicular. [3 marks]
   2. Explain briefly why lines \(l_1\) and \(l_2\) cannot be parallel. [2 marks]
   3. Given that \(\theta\) is the acute angle between lines \(l_1\) and \(l_2\), show that $$\cos \theta = \frac{p}{\sqrt{34n^2 + qn + 306}}$$ where \(p\) and \(q\) are constants to be found. [3 marks]

Line \(l_1\) has equation $$\frac{x - 2}{3} = \frac{1 - 2y}{4} = -z$$ and line \(l_2\) has equation $$\mathbf{r} = \begin{bmatrix} -7 \\ 4 \\ -2 \end{bmatrix} + \mu \begin{bmatrix} 12 \\ a + 3 \\ 2b \end{bmatrix}$$

1. In the case when \(l_1\) and \(l_2\) are parallel, show that \(a = -11\) and find the value of \(b\). [4 marks]
2. In a different case, the lines \(l_1\) and \(l_2\) intersect at exactly one point, and the value of \(b\) is 3 Find the value of \(a\). [5 marks]

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]

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]

The line \(L_1\) has equation $$\mathbf{r} = \begin{pmatrix} 4 \\ 2 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 3 \\ -1 \end{pmatrix}$$ The transformation T is represented by the matrix $$\begin{pmatrix} 2 & 1 & 0 \\ 3 & 4 & 6 \\ -5 & 2 & -3 \end{pmatrix}$$ The transformation T transforms the line \(L_1\) to the line \(L_2\)

1. Show that the angle between \(L_1\) and \(L_2\) is 0.701 radians, correct to three decimal places. [4 marks]
2. Find the shortest distance between \(L_1\) and \(L_2\) Give your answer in an exact form. [6 marks]