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

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 \(A(7, 2, 8)\), \(B(7, -4, 0)\) 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]

The line \(l_1\) passes through the points \(A(6, 2, 7)\) and \(B(4, -3, 7)\)

1. Find a Cartesian equation of \(l_1\) [2 marks]
2. The line \(l_2\) has vector equation \(\mathbf{r} = \begin{pmatrix} 8 \\ 9 \\ c \end{pmatrix} + \mu \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}\) where \(c\) is a constant.
   1. Explain how you know that the lines \(l_1\) and \(l_2\) are not perpendicular. [2 marks]
   2. The lines \(l_1\) and \(l_2\) both lie in the same plane. Find the value of \(c\) [5 marks]

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]

Find the acute angle between the lines with vector equations \(\mathbf{r} = \begin{pmatrix} 3 \\ 0 \\ -2 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}\) and \(\mathbf{r} = \begin{pmatrix} 1 \\ 5 \\ 3 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ 1 \\ -2 \end{pmatrix}\). [3]

The line \(L_1\) passes through the points \(A(1, 2, -3)\) and \(B(-2, 1, 0)\).

1. 1. Show that the vector equation of \(L_1\) can be written as $$\mathbf{r} = (1 - 3\lambda)\mathbf{i} + (2 - \lambda)\mathbf{j} + (-3 + 3\lambda)\mathbf{k}.$$
   2. Write down the equation of \(L_1\) in Cartesian form. [4]

The vector equation of the line \(L_2\) is given by \(\mathbf{r} = 2\mathbf{i} - 4\mathbf{j} + \mu(4\mathbf{j} + 7\mathbf{k})\).

1. Show that \(L_1\) and \(L_2\) do not intersect. [5]
2. Find a vector in the direction of the common perpendicular to \(L_1\) and \(L_2\). [5]

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]

Two lines, \(l_1\) and \(l_2\), have the following equations. $$l_1: \mathbf{r} = \begin{pmatrix} -11 \\ 10 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix}$$ $$l_2: \mathbf{r} = \begin{pmatrix} 5 \\ 2 \\ 4 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ 1 \\ -2 \end{pmatrix}$$ \(P\) is the point of intersection of \(l_1\) and \(l_2\).

1. Find the position vector of \(P\). [2]
2. Find, correct to 1 decimal place, the acute angle between \(l_1\) and \(l_2\). [2]

1. Find the value of \(k\) such that \(\begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}\) and \(\begin{pmatrix} -2 \\ 3 \\ k \end{pmatrix}\) are perpendicular. [2]

Two lines have equations \(l_1: \mathbf{r} = \begin{pmatrix} 3 \\ 2 \\ 7 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ -1 \\ 3 \end{pmatrix}\) and \(l_2: \mathbf{r} = \begin{pmatrix} 6 \\ 5 \\ 2 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix}\).

1. Find the point of intersection of \(l_1\) and \(l_2\). [4]
2. The vector \(\begin{pmatrix} 1 \\ a \\ b \end{pmatrix}\) is perpendicular to the lines \(l_1\) and \(l_2\). Find the values of \(a\) and \(b\). [5]

With respect to a fixed origin \(O\), the line \(l\) has equation $$\mathbf{r} = \begin{pmatrix} 13 \\ 8 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 2 \\ -1 \end{pmatrix}, \text{ where } \lambda \text{ is a scalar parameter.}$$ The point \(A\) lies on \(l\) and has coordinates \((3, -2, 6)\). The point \(P\) has position vector \((-\mathbf{i} + 2\mathbf{k})\) relative to \(O\). Given that vector \(\overrightarrow{PA}\) is perpendicular to \(l\), and that point \(B\) is a point on \(l\) such that \(\angle BPA = 45°\), find the coordinates of the two possible positions of \(B\). [5]

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]

With respect to a fixed origin \(O\), the lines \(l_1\) and \(l_2\) are given by the equations \begin{align} l_1: \quad \mathbf{r} &= (-9\mathbf{i} + 10\mathbf{k}) + \lambda(2\mathbf{i} + \mathbf{j} - \mathbf{k})
l_2: \quad \mathbf{r} &= (3\mathbf{i} + \mathbf{j} + 17\mathbf{k}) + \mu(3\mathbf{i} - \mathbf{j} + 5\mathbf{k}) \end{align} where \(\lambda\) and \(\mu\) are scalar parameters.

1. Show that \(l_1\) and \(l_2\) meet and find the position vector of their point of intersection. [6]
2. Show that \(l_1\) and \(l_2\) are perpendicular to each other. [2]

The point \(A\) has position vector \(5\mathbf{i} + 7\mathbf{j} + 3\mathbf{k}\).

1. Show that \(A\) lies on \(l_1\). [1]

Two lines, \(l_1\) and \(l_2\), have the following equations. $$l_1: \mathbf{r} = \begin{pmatrix} -1 \\ 10 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix}$$ $$l_2: \mathbf{r} = \begin{pmatrix} 5 \\ 2 \\ 4 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ 1 \\ -2 \end{pmatrix}$$ P is the point of intersection of \(l_1\) and \(l_2\).

1. Find the position vector of P. [3]
2. Find, correct to 1 decimal place, the acute angle between \(l_1\) and \(l_2\). [3]

Q is a point on \(l_1\) which is 12 metres away from P. R is the point on \(l_2\) such that QR is perpendicular to \(l_1\).

1. Determine the length QR. [2]

The points \(A\) and \(B\) have position vectors \(5\mathbf{j} + 11\mathbf{k}\) and \(c\mathbf{i} + d\mathbf{j} + 21\mathbf{k}\) respectively, where \(c\) and \(d\) are constants. The line \(l\), through the points \(A\) and \(B\), has vector equation \(\mathbf{r} = 5\mathbf{j} + 11\mathbf{k} + \lambda(2\mathbf{i} + \mathbf{j} + 5\mathbf{k})\), where \(\lambda\) is a parameter.

1. Find the value of \(c\) and the value of \(d\). [3]

The point \(P\) lies on the line \(l\), and \(\overrightarrow{OP}\) is perpendicular to \(l\), where \(O\) is the origin.

1. Find the position vector of \(P\). [6]
2. Find the area of triangle \(OAB\), giving your answer to 3 significant figures. [4]

With respect to a fixed origin \(O\), the lines \(l_1\) and \(l_2\) are given by the equations $$l_1: \mathbf{r} = (\mathbf{i} + 5\mathbf{j} + 5\mathbf{k}) + \lambda(2\mathbf{i} + \mathbf{j} - \mathbf{k})$$ $$l_2: \mathbf{r} = (2\mathbf{j} + 12\mathbf{k}) + \mu(3\mathbf{i} - \mathbf{j} + 5\mathbf{k})$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Show that \(l_1\) and \(l_2\) meet and find the position vector of their point of intersection. [6]
2. Show that \(l_1\) and \(l_2\) are perpendicular to each other. [2]

The point \(A\), with position vector \(5\mathbf{i} + 7\mathbf{j} + 3\mathbf{k}\), lies on \(l_1\) The point \(B\) is the image of \(A\) after reflection in the line \(l_2\)

1. Find the position vector of \(B\). [3]

The equations of two lines are \(\mathbf{r} = \mathbf{i} + 2\mathbf{j} + \lambda(2\mathbf{i} + \mathbf{j} + 3\mathbf{k})\) and \(\mathbf{r} = 6\mathbf{i} + 8\mathbf{j} + \mathbf{k} + \mu(\mathbf{i} + 4\mathbf{j} - 5\mathbf{k})\).

1. Show that these lines meet, and find the coordinates of the point of intersection. [5]
2. Find the acute angle between these lines. [3]

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]

1. 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]

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]

Two lines, \(l_1\) and \(l_2\), have the following equations. $$l_1: \mathbf{r} = \begin{pmatrix} -11 \\ 10 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix}$$ $$l_2: \mathbf{r} = \begin{pmatrix} 5 \\ 2 \\ 4 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ 1 \\ -2 \end{pmatrix}$$ \(P\) is the point of intersection of \(l_1\) and \(l_2\).

1. Find the position vector of \(P\). [3]
2. Find, correct to 1 decimal place, the acute angle between \(l_1\) and \(l_2\). [3]

\(Q\) is a point on \(l_1\) which is 12 metres away from \(P\). \(R\) is the point on \(l_2\) such that \(QR\) is perpendicular to \(l_1\).

1. Determine the length \(QR\). [2]

Line \(l_1\) has Cartesian equation $$l_1: \frac{-x}{2} = \frac{y-5}{2} = \frac{-z-6}{7}.$$

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

Line \(l_2\) has vector equation $$l_2: \mathbf{r} = \begin{pmatrix} 2 \\ 7 \\ -1 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ -2 \\ 4 \end{pmatrix}.$$

1. Find the point of intersection of \(l_1\) and \(l_2\). [3]
2. Find the acute angle between \(l_1\) and \(l_2\). [3]

The points \(A\) and \(B\) have position vectors \(\mathbf{i} - \mathbf{j} + \mathbf{k}\) and \(2\mathbf{i} + \mathbf{j} + 3\mathbf{k}\) respectively, relative to the origin \(O\). The point \(C\) is on the line \(OA\) extended so that \(\overrightarrow{AC} = 2\overrightarrow{OA}\) and the point \(D\) is on the line \(OB\) extended so that \(\overrightarrow{BD} = 3\overrightarrow{OB}\). The point \(X\) is such that \(OCXD\) is a parallelogram.

1. Show that a vector equation of the line \(AX\) is \(\mathbf{r} = \mathbf{i} - \mathbf{j} + \mathbf{k} + \lambda(5\mathbf{i} + 7\mathbf{k})\) and find an equation of the line \(CD\) in a similar form. [5]
2. Prove that the lines \(AX\) and \(CD\) intersect and find the position vector of their point of intersection. [4]