4.04c Scalar product: calculate and use for angles

8 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. \begin{figure}[h] \end{figure} Relative to axes \(\mathrm { O } x\) (due east), \(\mathrm { O } y\) (due north) and \(\mathrm { O } z\) (vertically upwards), the coordinates of the points are as follows.
A: (0, 0, -15)
B: (100, 0, -30)
C: (0, 100, -25)
D: (0, 0, -40)
E: (100, 0, -50)
F: (0, 100, -35)

1. Verify that the cartesian equation of the plane ABC is \(3 x + 2 y + 20 z + 300 = 0\).
2. Find the vectors \(\overrightarrow { \mathrm { DE } }\) and \(\overrightarrow { \mathrm { DF } }\). 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 .
3. By calculating the angle between their normal vectors, find the angle between the planes ABC and DEF. It is decided to drill down to the seam from a point \(\mathrm { R } ( 15,34,0 )\) in a line perpendicular to the upper surface of the seam. This line meets the plane ABC at the point S .
4. Write down a vector equation of the line RS. Calculate the coordinates of S.

1 The point A has coordinates \(( 2,5,4 )\) and the line BC has equation $$\mathbf { r } = \left( \begin{array} { c } 8 \\ 25 \\ 43 \end{array} \right) + \lambda \left( \begin{array} { c } 4 \\ 15 \\ 25 \end{array} \right)$$ You are given that \(\mathrm { AB } = \mathrm { AC } = 15\).

1. Show that the coordinates of one of the points B and C are (4, 10, 18). Find the coordinates of the other point. These points are B and C respectively.
2. Find the equation of the plane ABC in cartesian form.
3. Show that the plane containing the line BC and perpendicular to the plane ABC has equation \(5 y - 3 z + 4 = 0\). The point D has coordinates \(( 1,1,3 )\).
4. Show that \(| \overrightarrow { B C } \times \overrightarrow { A D } | = \sqrt { 7667 }\) and hence find the shortest distance between the lines \(B C\) and \(A D\).
5. Find the volume of the tetrahedron ABCD .

3.The lines \(L _ { 1 }\) and \(L _ { 2 }\) have the equations $$L _ { 1 } : \mathbf { r } = \left( \begin{array} { l } 1 \\ 0 \\ 9 \end{array} \right) + s \left( \begin{array} { l } 2 \\ p \\ 6 \end{array} \right) \quad \text { and } \quad L _ { 2 } : \mathbf { r } = \left( \begin{array} { r } - 15 \\ 12 \\ - 9 \end{array} \right) + t \left( \begin{array} { r } 4 \\ - 5 \\ 2 \end{array} \right)$$ where \(p\) is a constant.
The acute angle between \(L _ { 1 }\) and \(L _ { 2 }\) is \(\theta\) where \(\cos \theta = \frac { \sqrt { 5 } } { 3 }\)

1. Find the value of \(p\) . The line \(L _ { 3 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } - 15 \\ 12 \\ - 9 \end{array} \right) + u \left( \begin{array} { r } 8 \\ - 6 \\ - 5 \end{array} \right)\) and the lines \(L _ { 3 }\) and \(L _ { 2 }\) intersect at the point \(A\) .
   The point \(B\) on \(L _ { 2 }\) has position vector \(\left( \begin{array} { r } 5 \\ - 13 \\ 1 \end{array} \right)\) and point \(C\) lies on \(L _ { 3 }\) such that \(A B D C\) is a rhombus.
2. Find the two possible position vectors of \(D\) .

5 Show that the vectors \(\left[ \begin{array} { c } 1 \\ - 3 \\ 5 \end{array} \right]\) and \(\left[ \begin{array} { l } 7 \\ 4 \\ 1 \end{array} \right]\) are perpendicular.

15 Two submarines are travelling on different straight lines. The two lines are described by the equations $$\mathbf { r } = \left[ \begin{array} { c } 2 \\ - 1 \\ 4 \end{array} \right] + \lambda \left[ \begin{array} { c } 5 \\ 3 \\ - 2 \end{array} \right] \quad \text { and } \quad \frac { x - 5 } { 4 } = \frac { y } { 2 } = 4 - z$$ 15

1. 1. Show that the two lines intersect.
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      1. (ii) Find the position vector of the point of intersection.
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   2. Tracey says that the submarines will collide because there is a common point on the two lines. Explain why Tracey is not necessarily correct. 15
   3. Calculate the acute angle between the lines $$\mathbf { r } = \left[ \begin{array} { c } 2 \\ - 1 \\ 4 \end{array} \right] + \lambda \left[ \begin{array} { c } 5 \\ 3 \\ - 2 \end{array} \right] \quad \text { and } \quad \frac { x - 5 } { 4 } = \frac { y } { 2 } = 4 - z$$ Give your angle to the nearest \(0.1 ^ { \circ }\)

7 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left[ \begin{array} { c } 3 \\ 1 \\ - 2 \end{array} \right] + \lambda \left[ \begin{array} { c } 3 \\ - 4 \\ 1 \end{array} \right] \\ & l _ { 2 } : \mathbf { r } = \left[ \begin{array} { c } - 12 \\ a \\ - 3 \end{array} \right] + \mu \left[ \begin{array} { c } 3 \\ 2 \\ - 1 \end{array} \right] \end{aligned}$$ 7

1. Show that the point \(P ( - 3,9 , - 4 )\) lies on \(l _ { 1 }\) 7
2. Show that \(l _ { 1 }\) is perpendicular to \(l _ { 2 }\) 7
3. Given that the lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect, calculate the value of the constant \(a\) 7
4. Hence, find the coordinates of the point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\)

5 The vectors \(\mathbf { a }\) and \(\mathbf { b }\) are given by $$\mathbf { a } = 3 \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k } \quad \text { and } \quad \mathbf { b } = 2 \mathbf { i } - \mathbf { j } - 5 \mathbf { k }$$ 5

1. Calculate a.b 5
2. \(\quad\) Calculate \(| \mathbf { a } |\) and \(| \mathbf { b } |\) \(| \mathbf { a } | =\) \(\_\_\_\_\) 5
3. Calculate the acute angle between \(\mathbf { a }\) and \(\mathbf { b }\) Give your answer to the nearest degree.

7 A lighting engineer is setting up part of a display inside a large building. The diagram shows a plan view of the area in which he is working. He has two lights, which project narrow beams of light. One is set up at a point 3 metres above the point \(A\) and the beam from this light hits the wall 23 metres above the point \(D\). The other is set up 1 metre above the point \(B\) and the beam from this light hits the wall 29 metres above the point \(C\). \includegraphics[max width=\textwidth, alt={}, center]{e61d0202-49c9-4ed9-9fa3-f10734e17463-10_776_1301_826_392} 7

1. By creating a suitable model, show that the beams of light intersect. 7
2. Find the angle between the two beams of light.
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3. State one way in which the model you created in part (a) could be refined.
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3 The line \(L\) has equation \(\mathbf { r } = \left[ \begin{array} { l } 3 \\ 2 \\ 0 \end{array} \right] + \lambda \left[ \begin{array} { c } - 1 \\ - 2 \\ 5 \end{array} \right]\) Which of the following lines is perpendicular to the line \(L\) ?
Tick \(( \checkmark )\) one box. $$\begin{aligned} & \mathbf { r } = \left[ \begin{array} { c } 2 \\ - 3 \\ 4 \end{array} \right] + \mu \left[ \begin{array} { c } 1 \\ 2 \\ - 5 \end{array} \right] \\ & \mathbf { r } = \left[ \begin{array} { l } 1 \\ 0 \\ 1 \end{array} \right] + \mu \left[ \begin{array} { c } 2 \\ - 3 \\ 1 \end{array} \right] \\ & \mathbf { r } = \left[ \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right] + \mu \left[ \begin{array} { l } 1 \\ 1 \\ 2 \end{array} \right] \\ & \mathbf { r } = \left[ \begin{array} { l } 0 \\ 3 \\ 2 \end{array} \right] + \mu \left[ \begin{array} { l } 4 \\ 3 \\ 2 \end{array} \right] \end{aligned}$$ □
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4 The vector \(\mathbf { p }\), all of whose components are positive, is given by \(\mathbf { p } = \left( \begin{array} { c } a ^ { 2 } \\ a - 5 \\ 26 \end{array} \right)\) where \(a\) is a constant.
You are given that \(\mathbf { p }\) is perpendicular to the vector \(\left( \begin{array} { c } 2 \\ 6 \\ - 3 \end{array} \right)\).
Determine the value of \(a\).

3 The equations of two intersecting lines are \(\mathbf { r } = \left( \begin{array} { c } - 12 \\ a \\ - 1 \end{array} \right) + \lambda \left( \begin{array} { l } 2 \\ 2 \\ 1 \end{array} \right) \quad \mathbf { r } = \left( \begin{array} { l } 2 \\ 0 \\ 5 \end{array} \right) + \mu \left( \begin{array} { c } - 3 \\ 1 \\ - 1 \end{array} \right)\) where \(a\) is a constant.

1. Find a vector, \(\mathbf { b }\), which is perpendicular to both lines.
2. Show that b. \(\left( \begin{array} { c } - 12 \\ a \\ - 1 \end{array} \right) =\) b. \(\left( \begin{array} { l } 2 \\ 0 \\ 5 \end{array} \right)\).
3. Hence, or otherwise, find the value of \(a\).

The points \(A , B\) and \(C\) have position vectors \(\mathbf { i } , 2 \mathbf { j }\) and \(4 \mathbf { k }\) respectively, relative to an origin \(O\). The point \(N\) is the foot of the perpendicular from \(O\) to the plane \(A B C\). The point \(P\) on the line-segment \(O N\) is such that \(O P = \frac { 3 } { 4 } O N\). The line \(A P\) meets the plane \(O B C\) at \(Q\). Find a vector perpendicular to the plane \(A B C\) and show that the length of \(O N\) is \(\frac { 4 } { \sqrt { } ( 21 ) }\). Find the position vector of the point \(Q\). Show that the acute angle between the planes \(A B C\) and \(A B Q\) is \(\cos ^ { - 1 } \left( \frac { 2 } { 3 } \right)\).

9 Three points \(A , B\) and \(C\) have coordinates \(( 1,0,7 ) , ( 13,9,1 )\) and \(( 2 , - 1 , - 7 )\) respectively.

1. Use a scalar product to find angle \(A C B\).
2. Hence find the area of triangle \(A C B\).
3. Show that a vector equation of the line \(A B\) is given by \(\mathbf { r } = \mathbf { i } + 7 \mathbf { k } + \lambda ( 4 \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k } )\), where \(\lambda\) is a scalar parameter.

4 The points \(A , B , C\) and \(D\) have coordinates \(( 2 , - 1,0 ) , ( 3,2,5 ) , ( 4,2,3 )\) and \(( - 1 , a , b )\) respectively, where \(a\) and \(b\) are constants.

1. Find the angle \(A B C\).
2. Given that the lines \(A B\) and \(C D\) are parallel, find the values of \(a\) and \(b\).

7 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have the following vector equations. $$\begin{aligned} & l _ { 1 } : \mathbf { r } = 3 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } + \lambda ( \mathbf { i } - 6 \mathbf { j } - 2 \mathbf { k } ) \\ & l _ { 2 } : \mathbf { r } = \mathbf { i } + 5 \mathbf { j } + 2 \mathbf { k } + \mu ( 3 \mathbf { j } + \mathbf { k } ) \end{aligned}$$

1. Show that the lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect and find the coordinates of their point of intersection.
2. Find the acute angle between the lines \(l _ { 1 }\) and \(l _ { 2 }\).

8 The line \(l\) has equation \(\mathbf { r } = \lambda \mathbf { d }\) and the plane \(\Pi _ { 1 }\) has equation \(\mathbf { r } . \mathbf { n } = 35\), where $$\mathbf { d } = \left( \begin{array} { r } 2 \\ - 1 \\ 2 \end{array} \right) \quad \text { and } \quad \mathbf { n } = \left( \begin{array} { r } 6 \\ - 2 \\ 3 \end{array} \right) .$$

1. (a) Determine the exact value of \(\cos \theta\), where \(\theta\) is the angle between \(\mathbf { d }\) and \(\mathbf { n }\).
   (b) Determine the position vector of the point of intersection of \(l\) and \(\Pi _ { 1 }\).
   (c) Determine the shortest distance from \(O\) to \(\Pi _ { 1 }\).
2. The plane \(\Pi _ { 2 }\) has cartesian equation \(12 x - 4 y + 6 z + 21 = 0\). Determine the distance between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).

11 The points \(A\) and \(B\) have position vectors \(2 \mathbf { i } + \mathbf { j } - 3 \mathbf { k }\) and \(3 \mathbf { i } - 2 \mathbf { j } - \mathbf { k }\) respectively, relative to the origin \(O\). The point \(P\) lies on \(O A\) extended so that \(\overrightarrow { O P } = 3 \overrightarrow { O A }\) and the point \(Q\) lies on \(O B\) extended so that \(\overrightarrow { O Q } = 2 \overrightarrow { O B }\).

1. Find the coordinates of the point of intersection of the lines \(A Q\) and \(B P\).
2. Find the acute angle between the lines \(A Q\) and \(B P\).

6 Two straight lines have equations $$\mathbf { r } = - 3 \mathbf { i } + 11 \mathbf { j } - 9 \mathbf { k } + \lambda ( 4 \mathbf { i } + 7 \mathbf { j } + 8 \mathbf { k } )$$ and $$\mathbf { r } = 21 \mathbf { i } + 2 \mathbf { j } + 15 \mathbf { k } + \mu ( 3 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } )$$

1. Show that the lines intersect and find the coordinates of their point of intersection.
2. Find the acute angle between the two lines.

10 The line \(l _ { 1 }\) is parallel to the vector \(4 \mathbf { j } - \mathbf { k }\) and passes through the point \(A\) whose position vector is \(2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k }\). The variable line \(l _ { 2 }\) is parallel to the vector \(\mathbf { i } - ( 2 \sin t ) \mathbf { j }\), where \(0 \leqslant t < 2 \pi\), and passes through the point \(B\) whose position vector is \(\mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k }\). The points \(P\) and \(Q\) are on \(l _ { 1 }\) and \(l _ { 2 }\) respectively, and \(P Q\) is perpendicular to both \(l _ { 1 }\) and \(l _ { 2 }\).

1. Find the length of \(P Q\) in terms of \(t\).
2. Hence find the values of \(t\) for which \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
3. For the case \(t = \frac { 1 } { 4 } \pi\), find the perpendicular distance from \(A\) to the plane \(B P Q\), giving your answer correct to 3 decimal places.

The equations of two straight lines are $$\mathbf{r} = \mathbf{i} + \mathbf{j} + 2a\mathbf{k} + \lambda(3\mathbf{i} + 4\mathbf{j} + a\mathbf{k}) \quad \text{and} \quad \mathbf{r} = -3\mathbf{i} - \mathbf{j} + 4\mathbf{k} + \mu(-\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}),$$ where \(a\) is a constant.

1. Given that the acute angle between the directions of these lines is \(\frac{1}{4}\pi\), find the possible values of \(a\). [6]
2. Given instead that the lines intersect, find the value of \(a\) and the position vector of the point of intersection. [5]

Two lines have equations \(\mathbf{r} = \begin{pmatrix} 1 \\ 3 \\ 2 \end{pmatrix} + s \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}\) and \(\mathbf{r} = \begin{pmatrix} 2 \\ 1 \\ 4 \end{pmatrix} + t \begin{pmatrix} 1 \\ -1 \\ 4 \end{pmatrix}\).

1. Show that the lines are skew. [5]
2. Find the acute angle between the directions of the two lines. [3]

Referred to the origin \(O\), the points \(A\), \(B\) and \(C\) have position vectors given by $$\overrightarrow{OA} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k}, \quad \overrightarrow{OB} = 2\mathbf{i} + 4\mathbf{j} + \mathbf{k} \quad \text{and} \quad \overrightarrow{OC} = 3\mathbf{i} + 5\mathbf{j} - 3\mathbf{k}.$$

1. Find the exact value of the cosine of angle \(BAC\). [4]
2. Hence find the exact value of the area of triangle \(ABC\). [3]
3. Find the equation of the plane which is parallel to the \(y\)-axis and contains the line through \(B\) and \(C\). Give your answer in the form \(ax + by + cz = d\). [5]

Relative to the origin \(O\), the point \(A\) has position vector given by \(\overrightarrow{OA} = \mathbf{i} + 2\mathbf{j} + 4\mathbf{k}\). The line \(l\) has equation \(\mathbf{r} = 9\mathbf{i} - \mathbf{j} + 8\mathbf{k} + \mu(3\mathbf{i} - \mathbf{j} + 2\mathbf{k})\).

1. Find the position vector of the foot of the perpendicular from \(A\) to \(l\). Hence find the position vector of the reflection of \(A\) in \(l\). [5]
2. Find the equation of the plane through the origin which contains \(l\). Give your answer in the form \(ax + by + cz = d\). [3]
3. Find the exact value of the perpendicular distance of \(A\) from this plane. [3]

\includegraphics{figure_5} The diagram shows a three-dimensional shape. The base \(OAB\) is a horizontal triangle in which angle \(AOB\) is 90°. The side \(OBCD\) is a rectangle and the side \(OAD\) lies in a vertical plane. Unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are parallel to \(OA\) and \(OB\) respectively and the unit vector \(\mathbf{k}\) is vertical. The position vectors of \(A\), \(B\) and \(D\) are given by \(\overrightarrow{OA} = 8\mathbf{i}\), \(\overrightarrow{OB} = 5\mathbf{j}\) and \(\overrightarrow{OD} = 2\mathbf{i} + 4\mathbf{k}\).

1. Express each of the vectors \(\overrightarrow{DA}\) and \(\overrightarrow{CA}\) in terms of \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\). [2]
2. Use a scalar product to find angle \(CAD\). [4]

\includegraphics{figure_9} The diagram shows a pyramid \(OABCD\) with a horizontal rectangular base \(OABC\). The sides \(OA\) and \(AB\) have lengths of 8 units and 6 units respectively. The point \(E\) on \(OB\) is such that \(OE = 2\) units. The point \(D\) of the pyramid is 7 units vertically above \(E\). Unit vectors \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\) are parallel to \(OA\), \(OC\) and \(ED\) respectively.

1. Show that \(\overrightarrow{OE} = 1.6\mathbf{i} + 1.2\mathbf{j}\). [2]
2. Use a scalar product to find angle \(BDO\). [7]