4.04h Shortest distances: between parallel lines and between skew lines

11 The lines \(l _ { 1 } , l _ { 2 }\) and \(l _ { 3 }\) are defined as follows. $$\begin{aligned} & l _ { 1 } : \left( \mathbf { r } - \left[ \begin{array} { c } 1 \\ 5 \\ - 1 \end{array} \right] \right) \times \left[ \begin{array} { c } - 2 \\ 1 \\ - 3 \end{array} \right] = \mathbf { 0 } \\ & l _ { 2 } : \left( \mathbf { r } - \left[ \begin{array} { c } - 3 \\ 2 \\ 7 \end{array} \right] \right) \times \left[ \begin{array} { c } 2 \\ - 1 \\ 3 \end{array} \right] = \mathbf { 0 } \\ & l _ { 3 } : \left( \mathbf { r } - \left[ \begin{array} { c } - 5 \\ 12 \\ - 4 \end{array} \right] \right) \times \left[ \begin{array} { l } 4 \\ 0 \\ 9 \end{array} \right] = \mathbf { 0 } \end{aligned}$$ 11

1. 1. Explain how you know that two of the lines are parallel.
      

      11 (ii) Show that the perpendicular distance between these two parallel lines is 7.95 units, correct to three significant figures. [5 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

      11
2. Show that the lines \(l _ { 1 }\) and \(l _ { 3 }\) meet, and find the coordinates of their point of intersection. \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-23_2488_1716_219_153}

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 \(l\). \(l\) 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 \(\mathbf { b }\).
3. Write down an equation of \(l\) in vector form. \(P\) is a point on \(l\) such that \(P A = 2 O A\).
4. Find angle \(P O A\) giving your answer to 3 significant figures. \(C\) 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 \(p\).

9 The points \(P ( 3,5 , - 21 )\) and \(Q ( - 1,3 , - 16 )\) are on the ceiling of a long straight underground tunnel. A ventilation shaft must be dug from the point \(M\) on the ceiling of the tunnel midway between \(P\) and \(Q\) to horizontal ground level (where the \(z\)-coordinate is 0 ). The ventilation shaft must be perpendicular to the tunnel. The path of the ventilation shaft is modelled by the vector equation \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\), where \(\mathbf { a }\) is the position vector of \(M\). You are given that \(\mathbf { b } = \left( \begin{array} { l } 1 \\ \mathrm {~s} \\ \mathrm { t } \end{array} \right)\) where \(s\) and \(t\) are real numbers.

1. Show that \(\mathrm { S } = 2.5 \mathrm { t } - 2\).
2. Show that at the point where the ventilation shaft reaches the ground \(\lambda = \frac { \mathrm { C } } { \mathrm { t } }\), where \(c\) is a constant to be determined.
3. Using the results in parts (a) and (b), determine the shortest possible length of the ventilation shaft.
4. Explain what the fact that \(\mathbf { b } \times \left( \begin{array} { l } 0 \\ 0 \\ 1 \end{array} \right) \neq \mathbf { O }\) means about the direction of the ventilation shaft.

2 Find a vector which is perpendicular to both of the lines $$\mathbf { r } = \left( \begin{array} { r } 11 \\ 5 \\ 4 \end{array} \right) + \lambda \left( \begin{array} { l } 6 \\ 2 \\ 5 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { r } 1 \\ 7 \\ - 1 \end{array} \right) + \mu \left( \begin{array} { r } - 6 \\ 1 \\ 4 \end{array} \right)$$ and hence find the shortest distance between them.

The points \(A\) and \(B\) have position vectors, relative to the origin \(O\), given by $$\overrightarrow{OA} = \begin{pmatrix} -1 \\ 3 \\ 5 \end{pmatrix} \quad \text{and} \quad \overrightarrow{OB} = \begin{pmatrix} 3 \\ -1 \\ -4 \end{pmatrix}.$$ The line \(l\) passes through \(A\) and is parallel to \(OB\). The point \(N\) is the foot of the perpendicular from \(B\) to \(l\).

1. State a vector equation for the line \(l\). [1]
2. Find the position vector of \(N\) and show that \(BN = 3\). [6]
3. Find the equation of the plane containing \(A\), \(B\) and \(N\), giving your answer in the form \(ax + by + cz = d\). [5]

The lines \(l_1\) and \(l_2\) have equations \(\mathbf{r} = \mathbf{i} + 3\mathbf{j} - 2\mathbf{k} + \lambda(2\mathbf{i} + \mathbf{j} + \mathbf{k})\) and \(\mathbf{r} = \mathbf{i} - 2\mathbf{j} + 9\mathbf{k} + \mu(\mathbf{i} - 4\mathbf{j} + 2\mathbf{k})\) respectively. The plane \(\Pi_1\) contains \(l_1\) and is parallel to \(l_2\).

1. Find the equation of \(\Pi_1\), giving your answer in the form \(ax + by + cz = d\). [4]

The plane \(\Pi_2\) contains \(l_2\) and the point with coordinates \((2, -1, 7)\).

1. Find the acute angle between \(\Pi_1\) and \(\Pi_2\). [4]

The point \(P\) on \(l_1\) and the point \(Q\) on \(l_2\) are such that \(PQ\) is perpendicular to both \(l_1\) and \(l_2\).

1. Find a vector equation for \(PQ\). [7]

The line \(l_1\) passes through the point \(A\) with position vector \(\mathbf{i} - \mathbf{j} - 2\mathbf{k}\) and is parallel to the vector \(3\mathbf{i} - 4\mathbf{j} - 2\mathbf{k}\). The variable line \(l_2\) passes through the point \((1 + 5 \cos t)\mathbf{i} - (1 + 5 \sin t)\mathbf{j} - 14\mathbf{k}\), where \(0 \leq t < 2\pi\), and is parallel to the vector \(15\mathbf{i} + 8\mathbf{j} - 3\mathbf{k}\). The points \(P\) and \(Q\) are on \(l_1\) and \(l_2\) respectively, and \(PQ\) is perpendicular to both \(l_1\) and \(l_2\).

1. Find the length of \(PQ\) in terms of \(t\). [4]
2. Hence show that the lines \(l_1\) and \(l_2\) do not intersect, and find the maximum length of \(PQ\) as \(t\) varies. [3]
3. The plane \(\Pi_1\) contains \(l_1\) and \(PQ\); the plane \(\Pi_2\) contains \(l_2\) and \(PQ\). Find the angle between the planes \(\Pi_1\) and \(\Pi_2\), correct to the nearest tenth of a degree. [4]

The planes \(\Pi_1\) and \(\Pi_2\) have vector equations $$\mathbf{r} = \lambda_1(\mathbf{i} + \mathbf{j} - \mathbf{k}) + \mu_1(2\mathbf{i} - \mathbf{j} + \mathbf{k}) \quad \text{and} \quad \mathbf{r} = \lambda_2(\mathbf{i} + 2\mathbf{j} + \mathbf{k}) + \mu_2(3\mathbf{i} + \mathbf{j} - \mathbf{k})$$ respectively. The line \(l\) passes through the point with position vector \(4\mathbf{i} + 5\mathbf{j} + 6\mathbf{k}\) and is parallel to both \(\Pi_1\) and \(\Pi_2\). Find a vector equation for \(l\). [6] Find also the shortest distance between \(l\) and the line of intersection of \(\Pi_1\) and \(\Pi_2\). [4]

The position vectors of the points \(A, B, C, D\) are $$\mathbf{i} + \mathbf{j} + 3\mathbf{k}, \quad 3\mathbf{i} + 4\mathbf{j} + 5\mathbf{k}, \quad -\mathbf{i} + 3\mathbf{k}, \quad m\mathbf{j} + 4\mathbf{k},$$ respectively, where \(m\) is a constant.

1. Show that the lines \(AB\) and \(CD\) are parallel when \(m = \frac{3}{2}\). [1]
2. Given that \(m \neq \frac{3}{2}\), find the shortest distance between the lines \(AB\) and \(CD\). [5]
3. When \(m = 2\), find the acute angle between the planes \(ABC\) and \(ABD\), giving your answer in degrees. [6]

With \(O\) as the origin, the points \(A\), \(B\), \(C\) have position vectors $$\mathbf{i} - \mathbf{j}, \quad 2\mathbf{i} + \mathbf{j} + 7\mathbf{k}, \quad \mathbf{i} - \mathbf{j} + \mathbf{k}$$ respectively.

1. Find the shortest distance between the lines \(OC\) and \(AB\). [5]
2. Find the cartesian equation of the plane containing the line \(OC\) and the common perpendicular of the lines \(OC\) and \(AB\). [4]

With respect to a fixed origin \(O\), the lines \(l_1\) and \(l_2\) are given by the equations $$l_1: \mathbf{r} = \begin{pmatrix} 5 \\ -3 \\ p \end{pmatrix} + \lambda \begin{pmatrix} 0 \\ 1 \\ -3 \end{pmatrix}, \quad l_2: \mathbf{r} = \begin{pmatrix} 8 \\ 5 \\ -2 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ 4 \\ -5 \end{pmatrix}$$ where \(\lambda\) and \(\mu\) are scalar parameters and \(p\) is a constant. The lines \(l_1\) and \(l_2\) intersect at the point \(A\).

1. Find the coordinates of \(A\). [2]
2. Find the value of the constant \(p\). [3]
3. Find the acute angle between \(l_1\) and \(l_2\), giving your answer in degrees to 2 decimal places. [3]

The point \(B\) lies on \(l_2\) where \(\mu = 1\)

1. Find the shortest distance from the point \(B\) to the line \(l_1\), giving your answer to 3 significant figures. [3]

The position vectors of the points \(P\) and \(Q\) with respect to an origin \(O\) are \(5\mathbf{i} + 2\mathbf{j} - 9\mathbf{k}\) and \(4\mathbf{i} + 4\mathbf{j} - 6\mathbf{k}\) respectively.

1. Find a vector equation for the line \(PQ\). [2]

The position vector of the point \(T\) is \(\mathbf{i} + 2\mathbf{j} - \mathbf{k}\).

1. Write down a vector equation for the line \(OT\) and show that \(OT\) is perpendicular to \(PQ\). [4]

It is given that \(OT\) intersects \(PQ\).

1. Find the position vector of the point of intersection of \(OT\) and \(PQ\). [3]
2. Hence find the perpendicular distance from \(O\) to \(PQ\), giving your answer in an exact form. [2]

Two skew lines have equations $$\frac{x}{2} = \frac{y + 3}{1} = \frac{z - 6}{3} \quad \text{and} \quad \frac{x - 5}{3} = \frac{y + 1}{1} = \frac{z - 7}{5}.$$

1. Find the direction of the common perpendicular to the lines. [2]
2. Find the shortest distance between the lines. [4]

Find the perpendicular distance from the point with position vector \(12\mathbf{i} + 5\mathbf{j} + 3\mathbf{k}\) to the line with equation \(\mathbf{r} = \mathbf{i} + 2\mathbf{j} + 5\mathbf{k} + t(8\mathbf{i} + 3\mathbf{j} - 6\mathbf{k})\). [6]

The line \(l_1\) passes through the points \((0, 0, 10)\) and \((7, 0, 0)\) and the line \(l_2\) passes through the points \((4, 6, 0)\) and \((3, 3, 1)\). Find the shortest distance between \(l_1\) and \(l_2\). [7]

The points \(A(5, -4, 6)\) and \(B(6, -6, 8)\) lie on the line \(L\). The point \(C\) is \((15, -5, 9)\).

1. \(D\) is the point on \(L\) that is closest to \(C\). Find the coordinates of \(D\). [6 marks]
2. Hence find, in exact form, the shortest distance from \(C\) to \(L\). [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]

The line \(L_1\) has equation $$\frac{x-2}{3} = \frac{y+4}{8} = \frac{4z-5}{5}$$ The line \(L_2\) has equation $$\left(\mathbf{r} - \begin{bmatrix} -2 \\ 0 \\ 3 \end{bmatrix}\right) \times \begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix} = \mathbf{0}$$ Find the shortest distance between the two lines, giving your answer to three significant figures. [8 marks]

1. Find the shortest distance between the point \((-6, 4)\) and the line \(y = -0.75x + 7\). [2]

Two lines, \(l_1\) and \(l_2\), are given by $$l_1: \mathbf{r} = \begin{pmatrix} 4 \\ 3 \\ -2 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 1 \\ -4 \end{pmatrix} \text{ and } l_2: \mathbf{r} = \begin{pmatrix} 11 \\ -1 \\ 5 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ -1 \\ 1 \end{pmatrix}$$

1. Find the shortest distance between \(l_1\) and \(l_2\). [3]
2. Hence determine the geometrical arrangement of \(l_1\) and \(l_2\). [2]

1. Find the shortest distance between the point \((-6, 4)\) and the line \(y = -0.75x + 7\). [2]

Two lines, \(l_1\) and \(l_2\), are given by $$l_1: \mathbf{r} = \begin{pmatrix} 4 \\ 3 \\ -2 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 1 \\ -4 \end{pmatrix} \text{ and } l_2: \mathbf{r} = \begin{pmatrix} 11 \\ -1 \\ 5 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ -1 \\ 1 \end{pmatrix}.$$

1. Find the shortest distance between \(l_1\) and \(l_2\). [3]
2. Hence determine the geometrical arrangement of \(l_1\) and \(l_2\). [2]