4.04c Scalar product: calculate and use for angles

The position vectors of three points \(A\), \(B\) and \(C\) relative to an origin \(O\) are given respectively by $$\overrightarrow{OA} = 7\mathbf{i} + 3\mathbf{j} - 3\mathbf{k},$$ $$\overrightarrow{OB} = 4\mathbf{i} + 2\mathbf{j} - 4\mathbf{k}$$ and $$\overrightarrow{OC} = 5\mathbf{i} + 4\mathbf{j} - 5\mathbf{k}.$$

1. Find the angle between \(AB\) and \(AC\). [6]
2. Find the area of triangle \(ABC\). [2]

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]

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]

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, \(O\), the points \(A\) and \(B\) have position vectors \(\begin{pmatrix} 1 \\ 5 \\ -1 \end{pmatrix}\) and \(\begin{pmatrix} 6 \\ 3 \\ -6 \end{pmatrix}\) respectively. Find, in exact, simplified form,

1. the cosine of \(\angle AOB\), [4]
2. the area of triangle \(OAB\), [4]
3. the shortest distance from \(A\) to the line \(OB\). [2]

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]

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]

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]

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 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]

\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 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]

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]

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]

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 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]

Vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\), are given by \(\mathbf{a} = \mathbf{i} + (1-p)\mathbf{j} + (p+2)\mathbf{k}\), \(\mathbf{b} = 2\mathbf{i} + \mathbf{j} + \mathbf{k}\) and \(\mathbf{c} = \mathbf{i} + 14\mathbf{j} + (p-3)\mathbf{k}\) where \(p\) is a constant. You are given that \(\mathbf{a} \times \mathbf{b}\) is perpendicular to \(\mathbf{c}\). Determine the possible values of \(p\). [6]