4.04f Line-plane intersection: find point

Show that the straight lines with equations \(\mathbf{r} = \begin{pmatrix} 4 \\ 2 \\ 4 \end{pmatrix} + \lambda \begin{pmatrix} 3 \\ 0 \\ 1 \end{pmatrix}\) and \(\mathbf{r} = \begin{pmatrix} -1 \\ 4 \\ 9 \end{pmatrix} + \mu \begin{pmatrix} -1 \\ 1 \\ 3 \end{pmatrix}\) meet. Find their point of intersection. [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]

A line \(l\) has equation \(\mathbf{r} = 3\mathbf{i} + \mathbf{j} - 2\mathbf{k} + t(\mathbf{i} + 4\mathbf{j} + 2\mathbf{k})\) and a plane \(\Pi\) has equation \(8x - 7y + 10z = 7\). Determine whether \(l\) lies in \(\Pi\), is parallel to \(\Pi\) without intersecting it, or intersects \(\Pi\) at one point. [5]

\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]

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]

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 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 \(2\mathbf{i} + 3\mathbf{j} - 4\mathbf{k}\) 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 \(\Pi_1\) 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]

Relative to an origin \(O\), the points \(P\), \(Q\) and \(R\) have position vectors $$\mathbf{p} = \mathbf{i} + 2\mathbf{j} - 7\mathbf{k}, \quad \mathbf{q} = -3\mathbf{i} + 4\mathbf{j} + \mathbf{k} \quad \text{and} \quad \mathbf{r} = 6\mathbf{i} + 4\mathbf{j} + \alpha\mathbf{k}$$ respectively.

1. Determine \(\mathbf{p} \times \mathbf{q}\). [2]
2. Deduce the value of \(\alpha\) for which
   1. \(OR\) is normal to the plane \(OPQ\), [1]
   2. the volume of tetrahedron \(OPQR\) is 50, [3]
   3. \(R\) lies in the plane \(OPQ\). [2]