Vectors 3D & Lines

The plane \(x + 2y + cz = 4\) is perpendicular to the plane \(2x - cy + 6z = 9\), where \(c\) is a constant. Find the value of \(c\). [3]

2 The two vectors \(\mathbf { a }\) and \(\mathbf { b }\) are such that \(\mathbf { a } \cdot \mathbf { b } = 0\) State the angle between the vectors \(\mathbf { a }\) and \(\mathbf { b }\) Circle your answer.
[0pt] [1 mark] \(0 ^ { \circ } 45 ^ { \circ } 90 ^ { \circ } 180 ^ { \circ }\)

2 Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 3 \\ - 6 \\ p \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { r } 2 \\ - 6 \\ - 7 \end{array} \right)$$ and angle \(A O B = 90 ^ { \circ }\).

1. Find the value of \(p\). \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) The point \(C\) is such that \(\overrightarrow { O C } = \frac { 2 } { 3 } \overrightarrow { O A }\).
2. Find the unit vector in the direction of \(\overrightarrow { B C }\). \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

4 Show that the straight lines with equations \(\mathbf { r } = \begin{array} { r r r } 2 & + \lambda & 0 \\ 4 & & 1 \end{array}\) and \(\mathbf { r } = \quad + \mu \quad\) meet.
Find their point of intersection.

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]

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]

\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]

\includegraphics{figure_6} The diagram shows a solid figure \(ABCDEF\) in which the horizontal base \(ABC\) is a triangle right-angled at \(A\). The lengths of \(AB\) and \(AC\) are 8 units and 4 units respectively and \(M\) is the mid-point of \(AB\). The point \(D\) is 7 units vertically above \(A\). Triangle \(DEF\) lies in a horizontal plane with \(DE\), \(DF\) and \(FE\) parallel to \(AB\), \(AC\) and \(CB\) respectively and \(N\) is the mid-point of \(FE\). The lengths of \(DE\) and \(DF\) are 4 units and 2 units respectively. Unit vectors \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\) are parallel to \(\overrightarrow{AB}\), \(\overrightarrow{AC}\) and \(\overrightarrow{AD}\) respectively.

1. Find \(\overrightarrow{MF}\) in terms of \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\). [1]
2. Find \(\overrightarrow{FN}\) in terms of \(\mathbf{i}\) and \(\mathbf{j}\). [1]
3. Find \(\overrightarrow{MN}\) in terms of \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\). [1]
4. Use a scalar product to find angle \(FMN\). [4]

The square-based pyramid \(P\) has vertices \(A, B, C, D\) and \(E\). The position vectors of \(A, B, C\) and \(D\) are \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{c}\) and \(\mathbf{d}\) respectively where $$\mathbf{a} = \begin{pmatrix} -2 \\ 3 \\ -1 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 5 \\ 8 \\ -6 \end{pmatrix}, \quad \mathbf{c} = \begin{pmatrix} 2 \\ 5 \\ 3 \end{pmatrix}, \quad \mathbf{d} = \begin{pmatrix} 6 \\ 1 \\ 1 \end{pmatrix}$$

1. Find the vectors \(\overrightarrow{AB}\), \(\overrightarrow{AC}\), \(\overrightarrow{AD}\), \(\overrightarrow{BC}\), \(\overrightarrow{BD}\) and \(\overrightarrow{CD}\). [3]
2. Find
   1. the length of a side of the square base of \(P\),
   2. the cosine of the angle between one of the slanting edges of \(P\) and its base,
   3. the height of \(P\),
   4. the position vector of \(E\).
   [9] A second pyramid, identical to \(P\), is attached by its square base to the base of \(P\) to form an octahedron.
3. Find the position vector of the other vertex of this octahedron. [3]

1 Find the angle between the vectors \(\mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k }\) and \(2 \mathbf { i } + \mathbf { j } + \mathbf { k }\).

Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are \(\mathbf{3i} + 4\mathbf{j} - \mathbf{k}\) and \(5\mathbf{i} - 2\mathbf{j} - 3\mathbf{k}\) respectively.

1. Use a scalar product to find angle \(BOA\). [4]

The point \(C\) is the mid-point of \(AB\). The point \(D\) is such that \(\overrightarrow{OD} = 2\overrightarrow{OB}\).

1. Find \(\overrightarrow{DC}\). [4]

Relative to a fixed origin \(O\), the position vectors of the points \(A\), \(B\) and \(C\) are $$\overrightarrow{OA} = -3\mathbf{i} + \mathbf{j} - 9\mathbf{k}, \quad \overrightarrow{OB} = \mathbf{i} - \mathbf{k}, \quad \overrightarrow{OC} = 5\mathbf{i} + 2\mathbf{j} - 5\mathbf{k} \text{ respectively}.$$

1. Find the cosine of angle \(ABC\). [4]

The line \(L\) is the angle bisector of angle \(ABC\).

1. Show that an equation of \(L\) is \(\mathbf{r} = \mathbf{i} - \mathbf{k} + t(\mathbf{i} + 2\mathbf{j} - 7\mathbf{k})\). [4]
2. Show that \(|\overrightarrow{AB}| = |\overrightarrow{AC}|\). [2]

The circle \(S\) lies inside triangle \(ABC\) and each side of the triangle is a tangent to \(S\).

1. Find the position vector of the centre of \(S\). [7]
2. Find the radius of \(S\). [5]

4
- 3
12 \end{array} \right) + \mu \left( \begin{array} { r }

With respect to a fixed origin \(O\), the line \(l_1\) has vector equation $$\mathbf{r} = \begin{pmatrix} -9 \\ 8 \\ 5 \end{pmatrix} + \mu \begin{pmatrix} 5 \\ -4 \\ -3 \end{pmatrix}$$ where \(\mu\) is a scalar parameter. The point \(A\) is on \(l_1\) where \(\mu = 2\).

1. Write down the coordinates of \(A\). [1] The acute angle between \(OA\) and \(l_1\) is \(\theta\), where \(O\) is the origin.
2. Find the value of \(\cos \theta\). [3] The point \(B\) is such that \(\overrightarrow{OB} = 3\overrightarrow{OA}\). The line \(l_2\) passes through the point \(B\) and is parallel to the line \(l_1\).
3. Find a vector equation of \(l_2\). [2]
4. Find the length of \(OB\), giving your answer as a simplified surd. [1] The point \(X\) lies on \(l_2\). Given that the vector \(\overrightarrow{OX}\) is perpendicular to \(l_2\),
5. find the length of \(OX\), giving your answer to 3 significant figures. [3]

8. The points \(A\) and \(B\) have coordinates \(( 3,9 , - 7 )\) and \(( 13 , - 6 , - 2 )\) respectively.

1. Find, in vector form, an equation for the line \(l\) which passes through \(A\) and \(B\).
2. Show that the point \(C\) with coordinates \(( 9,0 , - 4 )\) lies on \(l\). The point \(D\) is the point on \(l\) closest to the origin, \(O\).
3. Find the coordinates of \(D\).
4. Find the area of triangle \(O A B\) to 3 significant figures.

Verify that the vector \(2\mathbf{i} - \mathbf{j} + 4\mathbf{k}\) is perpendicular to the plane through the points A(2, 0, 1), B(1, 2, 2) and C(0, -4, 1). Hence find the cartesian equation of the plane. [5]

Verify that the vector \(2\mathbf{i} - \mathbf{j} + 4\mathbf{k}\) is perpendicular to the plane through the points A(2, 0, 1), B(1, 2, 2) and C(0, -4, 1). Hence find the cartesian equation of the plane. [5]

Three planes have equations \begin{align} (4k + 1)x - 3y + (k - 5)z &= 3
(k - 1)x + (3 - k)y + 2z &= 1
7x - 3y + 4z &= 2 \end{align}

1. The planes do not meet at a unique point. Show that \(k = 4.5\) is one possible value of \(k\), and find the other possible value of \(k\). [3 marks]
2. For each value of \(k\) found in part (a), identify the configuration of the given planes. In each case fully justify your answer, stating whether or not the equations of the planes form a consistent system. [4 marks]

5 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\mathbf { r } = \left( \begin{array} { l } 4 \\ 6 \\ 4 \end{array} \right)$$

5 Verify that the point \(( - 1,6,5 )\) lies on both the lines $$\mathbf { r } = \left( \begin{array} { r } 1 \\ 2 \\ - 1 \end{array} \right) + \lambda \left( \begin{array} { r } - 1 \\ 2 \\ 3 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { l } 0 \\ 6 \\ 3 \end{array} \right) + \mu \left( \begin{array} { l } - 1 \\ 0 \\ 1 \end{array} \right)$$

1. The line \(l _ { 1 }\) has vector equation \(\mathbf { r } = \left( \begin{array} { r } 5 \\ - 2 \\ 4 \end{array} \right) + \lambda \left( \begin{array} { r } 6 \\ 3 \\ - 1 \end{array} \right)\), where \(\lambda\) is a scalar parameter. The line \(l _ { 2 }\) has vector equation \(\mathbf { r } = \left( \begin{array} { r } 10 \\ 5 \\ - 3 \end{array} \right) + \mu \left( \begin{array} { l } 3 \\ 1 \\ 2 \end{array} \right)\), where \(\mu\) is a scalar parameter.

Justify, giving reasons in each case, whether the lines \(l _ { 1 }\) and \(l _ { 2 }\) are parallel, intersecting or skew.
(6)

4. Relative to a fixed origin, two lines have the equations $$\begin{aligned} & \mathbf { r } = \left( \begin{array} { c } 7 \\ 0 \\ - 3 \end{array} \right) + \lambda \left( \begin{array} { c } 5 \\ 4 \\ - 2 \end{array} \right) \end{aligned}$$

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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7 \includegraphics[max width=\textwidth, alt={}, center]{1a4ddaa9-1ec2-4138-bfcb-a482fe6c942f-3_394_750_260_699} The diagram shows a trapezium \(A B C D\) in which \(B A\) is parallel to \(C D\). The position vectors of \(A , B\) and \(C\) relative to an origin \(O\) are given by $$\overrightarrow { O A } = \left( \begin{array} { l } 3 \\ 4 \\ 0 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { l } 1 \\ 3 \\ 2 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { l } 4 \\ 5 \\ 6 \end{array} \right)$$

1. Use a scalar product to show that \(A B\) is perpendicular to \(B C\).
2. Given that the length of \(C D\) is 12 units, find the position vector of \(D\).

5 The points A , B and C have coordinates \(( 2,0 , - 1 ) , ( 4,3 , - 6 )\) and \(( 9,3 , - 4 )\) respectively.

1. Show that AB is perpendicular to BC .
2. Find the area of triangle ABC .

6 Relative to an origin \(O\), the position vector of \(A\) is \(3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }\) and the position vector of \(B\) is \(7 \mathbf { i } - 3 \mathbf { j } + \mathbf { k }\).

1. Show that angle \(O A B\) is a right angle.
2. Find the area of triangle \(O A B\).

1. Relative to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations

$$l _ { 1 } : \mathbf { r } = \left( \begin{array} { l } 2 \\ 0 \\ 7 \end{array} \right) + \lambda \left( \begin{array} { r } 2 \\ - 2 \\ 1 \end{array} \right) \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 2 \\ 0 \\ 7 \end{array} \right) + \mu \left( \begin{array} { l } 8 \\ 4 \\ 1 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters.
The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(A\).

1. Write down the coordinates of \(A\). Given that the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\) is \(\theta\),
2. show that \(\sin \theta = k \sqrt { 2 }\), where \(k\) is a rational number to be found. The point \(B\) lies on \(l _ { 1 }\) where \(\lambda = 4\) The point \(C\) lies on \(l _ { 2 }\) such that \(A C = 2 A B\).
3. Find the exact area of triangle \(A B C\).
4. Find the coordinates of the two possible positions of \(C\).
   

3 The points \(A\) and \(B\) have position vectors \(\mathbf { a } = \left( \begin{array} { r } 3 \\ 2 \\ - 1 \end{array} \right)\) and \(\mathbf { b } = \left( \begin{array} { r } - 1 \\ 4 \\ 8 \end{array} \right)\) respectively.
Show that the exact value of the distance \(A B\) is \(\sqrt { \mathbf { 1 0 1 } }\).

1. Relative to a fixed origin \(O\)

• the point \(A\) has position vector \(5 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }\)
• the point \(B\) has position vector \(2 \mathbf { i } + 4 \mathbf { j } + a \mathbf { k }\) where \(a\) is a positive integer.
  1. Show that \(| \overrightarrow { O A } | = \sqrt { 38 }\)
  2. Find the smallest value of \(a\) for which

$$| \overrightarrow { O B } | > | \overrightarrow { O A } |$$

7. The rate of increase of the number, \(N\), of fish in a lake is modelled by the differential equation $$\frac { \mathrm { d } N } { \mathrm {~d} t } = \frac { ( k t - 1 ) ( 5000 - N ) } { t } \quad t > 0 , \quad 0 < N < 5000$$ In the given equation, the time \(t\) is measured in years from the start of January 2000 and \(k\) is a positive constant.

1. By solving the differential equation, show that $$N = 5000 - A t \mathrm { e } ^ { - k t }$$ where \(A\) is a positive constant. After one year, at the start of January 2001, there are 1200 fish in the lake. After two years, at the start of January 2002, there are 1800 fish in the lake.
2. Find the exact value of the constant \(A\) and the exact value of the constant \(k\).
3. Hence find the number of fish in the lake after five years. Give your answer to the nearest hundred fish. \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \section*{Question 7 continued} \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

8 The points \(A ( 3,2,1 ) , B ( 5,4 , - 3 ) , C ( 3,17 , - 4 )\) and \(D ( 1,6,3 )\) form a quadrilateral \(A B C D\).

1. Show that \(A B = A D\).
2. Find a vector equation of the line through \(A\) and the mid-point of \(B D\).
3. Show that \(C\) lies on the line found in part (ii).
4. What type of quadrilateral is \(A B C D\) ?

Relative to a fixed origin \(O\),
the point \(A\) has position vector \(\mathbf { i } + 7 \mathbf { j } - 2 \mathbf { k }\),
the point \(B\) has position vector \(4 \mathbf { i } + 3 \mathbf { j } + 3 \mathbf { k }\),
and the point \(C\) has position vector \(2 \mathbf { i } + 10 \mathbf { j } + 9 \mathbf { k }\).
Given that \(A B C D\) is a parallelogram,

1. find the position vector of point \(D\). The vector \(\overrightarrow { A X }\) has the same direction as \(\overrightarrow { A B }\).
   Given that \(| \overrightarrow { A X } | = 10 \sqrt { 2 }\),
2. find the position vector of \(X\).

9 \includegraphics[max width=\textwidth, alt={}, center]{518bb805-5b14-4b41-94fd-38a31a90c218-16_533_601_258_772} The diagram shows a trapezium \(O A B C\) in which \(O A\) is parallel to \(C B\). The position vectors of \(A\) and \(B\) relative to the origin \(O\) are given by \(\overrightarrow { O A } = \left( \begin{array} { r } 2 \\ - 2 \\ - 1 \end{array} \right)\) and \(\overrightarrow { O B } = \left( \begin{array} { l } 6 \\ 1 \\ 1 \end{array} \right)\).

1. Show that angle \(O A B\) is \(90 ^ { \circ }\).
   The magnitude of \(\overrightarrow { C B }\) is three times the magnitude of \(\overrightarrow { O A }\).
2. Find the position vector of \(C\).
3. Find the exact area of the trapezium \(O A B C\), giving your answer in the form \(a \sqrt { } b\), where \(a\) and \(b\) are integers.

6. Relative to a fixed origin \(O\), the points \(A\), \(B\) and \(C\) have coordinates ( \(2,1,9 ) , ( 5,2,7 )\) and \(( 4 , - 3,3 )\) respectively. The line \(l\) passes through the points \(A\) and \(B\).

1. Find a vector equation for the line \(l\).
2. Find, in degrees, the acute angle between the line \(I\) and the line \(A C\). The point \(D\) lies on the line \(l\) such that angle \(A C D\) is \(90 ^ { \circ }\)
3. Find the coordinates of \(D\).
4. Find the exact area of triangle \(A D C\), giving your answer as a fully simplified surd.

Relative to a fixed origin \(O\), the point \(A\) has position vector \(\mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k }\) and the point \(B\) has position vector \(- 2 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }\). The points \(A\) and \(B\) lie on a straight line \(l\).

1. Find \(\overrightarrow { A B }\).
2. Find a vector equation of \(l\). The point \(C\) has position vector \(2 \mathbf { i } + p \mathbf { j } - 4 \mathbf { k }\) with respect to \(O\), where \(p\) is a constant. Given that \(A C\) is perpendicular to \(l\), find
3. the value of \(p\),
4. the distance \(A C\).
   

6 The points \(A\) and \(B\) have coordinates \(( 3,2,10 )\) and \(( 5 , - 2,4 )\) respectively.
The line \(l\) passes through \(A\) and has equation \(\mathbf { r } = \left[ \begin{array} { r } 3 \\ 2 \\ 10 \end{array} \right] + \lambda \left[ \begin{array} { r } 3 \\ 1 \\ - 2 \end{array} \right]\).

1. Find the acute angle between \(l\) and the line \(A B\).
2. The point \(C\) lies on \(l\) such that angle \(A B C\) is \(90 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{fdd3905e-11f7-4b20-adfe-4c686018a221-12_360_339_762_852} Find the coordinates of \(C\).
3. The point \(D\) is such that \(B D\) is parallel to \(A C\) and angle \(B C D\) is \(90 ^ { \circ }\). The point \(E\) lies on the line through \(B\) and \(D\) and is such that the length of \(D E\) is half that of \(A C\). Find the coordinates of the two possible positions of \(E\).
   [0pt] [4 marks]

At time \(t = 0\) s, the position vector of an object \(A\) is \(\mathbf{i}\) m and the position vector of another object \(B\) is \(3\mathbf{i}\) m. The constant velocity vector of \(A\) is \(2\mathbf{i} + 5\mathbf{j} - 4k\) ms\(^{-1}\) and the constant velocity vector of \(B\) is \(\mathbf{i} + 3\mathbf{j} - 5k\) ms\(^{-1}\). Determine the value of \(t\) when \(A\) and \(B\) are closest together and find the least distance between \(A\) and \(B\). [9]

3 The map of a large area of open land is marked in 1 km squares and a point near the middle of the area is defined to be the origin. The vectors \(\binom { 1 } { 0 }\) and \(\binom { 0 } { 1 }\) are in the directions east and north. At time \(t\) hours the position vectors of two hikers, Ashok and Kumar, are given by: $$\begin{array} { l l } \text { Ashok } & \mathbf { r } _ { \mathrm { A } } = \binom { - 2 } { 0 } + \binom { 8 } { 1 } t , \\ \text { Kumar } & \mathbf { r } _ { \mathrm { K } } = \binom { 7 t } { 10 - 4 t } . \end{array}$$

1. Prove that the two hikers meet and give the coordinates of the point where this happens.
2. Compare the speeds of the two hikers.

1. An aeroplane leaves a runway and moves with a constant speed of \(V \mathrm {~km} / \mathrm { h }\) due north along a straight path inclined at an angle \(\arctan \left( \frac { 3 } { 4 } \right)\) to the horizontal.

A light aircraft is moving due north in a straight horizontal line in the same vertical plane as the aeroplane, at a height of 3 km above the runway. The light aircraft is travelling with a constant speed of \(2 V \mathrm {~km} / \mathrm { h }\).
At the moment the aeroplane leaves the runway, the light aircraft is at a horizontal distance \(d \mathrm {~km}\) behind the aeroplane. Both aircraft continue to move with the same trajectories due north.

1. Show that the distance, \(D \mathrm {~km}\), between the two aircraft \(t\) hours after the aeroplane leaves the runway satisfies $$D ^ { 2 } = \left( \frac { 6 } { 5 } V t - d \right) ^ { 2 } + \left( \frac { 3 } { 5 } V t - 3 \right) ^ { 2 }$$ Given that the distance between the two aircraft is never less than 2 km ,
2. find the range of possible values for \(d\).

3 Show that points \(\mathrm { A } ( 1,4,9 ) , \mathrm { B } ( 0,11,17 )\) and \(\mathrm { C } ( 3 , - 10 , - 7 )\) are collinear.

3 Show that points \(\mathrm { A } ( 1,4,9 ) , \mathrm { B } ( 0,11,17 )\) and \(\mathrm { C } ( 3 , - 10 , - 7 )\) are collinear.

7 Relative to an origin \(O\), the position vectors of points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 0 \\ 2 \\ - 3 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 2 \\ 5 \\ - 2 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { l } 3 \\ p \\ q \end{array} \right)$$

1. In the case where \(A B C\) is a straight line, find the values of \(p\) and \(q\).
2. In the case where angle \(B A C\) is \(90 ^ { \circ }\), express \(q\) in terms of \(p\).
3. In the case where \(p = 3\) and the lengths of \(A B\) and \(A C\) are equal, find the possible values of \(q\).

7 The equation of a straight line \(l\) is $$\mathbf { r } = \left( \begin{array} { l } 1 \\ 0 \\ 2 \end{array} \right) + t \left( \begin{array} { r } 1 \\ - 1 \\ 0 \end{array} \right) .$$ \(O\) is the origin.

1. Find the position vector of the point \(P\) on \(l\) such that \(O P\) is perpendicular to \(l\).
2. A point \(Q\) on \(l\) is such that the length of \(O Q\) is 3 units. Find the two possible position vectors of \(Q\). [3]

1. Relative to a fixed origin \(O\),
   the point \(A\) has position vector \(( 2 \mathbf { i } - 3 \mathbf { j } - 2 \mathbf { k } )\) the point \(B\) has position vector \(( 3 \mathbf { i } + 2 \mathbf { j } + 5 \mathbf { k } )\) the point \(C\) has position vector ( \(2 \mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k }\) )

The line \(l\) passes through the points \(A\) and \(B\).

1. Find the vector \(\overrightarrow { A B }\).
2. Find a vector equation for the line \(l\).
3. Show that the size of the angle \(C A B\) is \(62.8 ^ { \circ }\), to one decimal place.
4. Hence find the area of triangle \(C A B\), giving your answer to 3 significant figures. The point \(D\) lies on the line \(l\). Given that the area of triangle \(C A D\) is twice the area of triangle \(C A B\),
5. find the two possible position vectors of point \(D\).
   

14. Relative to a fixed origin \(O\), the line \(l\) has vector equation $$\mathbf { r } = \left( \begin{array} { r } - 1 \\ - 4 \\ 6 \end{array} \right) + \lambda \left( \begin{array} { r } 2 \\ 1 \\ - 1 \end{array} \right)$$ where \(\lambda\) is a scalar parameter. Points \(A\) and \(B\) lie on the line \(l\), where \(A\) has coordinates ( \(1 , a , 5\) ) and \(B\) has coordinates ( \(b , - 1,3\) ).

1. Find the value of the constant \(a\) and the value of the constant \(b\).
2. Find the vector \(\overrightarrow { A B }\). The point \(C\) has coordinates ( \(4 , - 3,2\) )
3. Show that the size of the angle \(C A B\) is \(30 ^ { \circ }\)
4. Find the exact area of the triangle \(C A B\), giving your answer in the form \(k \sqrt { 3 }\), where \(k\) is a constant to be determined. The point \(D\) lies on the line \(l\) so that the area of the triangle \(C A D\) is twice the area of the triangle \(C A B\).
5. Find the coordinates of the two possible positions of \(D\).

7 The position vectors of the points \(A\) and \(B\), relative to an origin \(O\), are given by $$\overrightarrow { O A } = \left( \begin{array} { l } 1 \\ 0 \\ 2 \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { r } k \\ - k \\ 2 k \end{array} \right)$$ where \(k\) is a constant.

1. In the case where \(k = 2\), calculate angle \(A O B\).
2. Find the values of \(k\) for which \(\overrightarrow { A B }\) is a unit vector.

The line \(L_1\) passes through the points \((2, -3, 1)\) and \((-1, -2, -4)\). The line \(L_2\) passes through the point \((3, 2, -9)\) and is parallel to the vector \(\mathbf{4i} - \mathbf{4j} + \mathbf{5k}\).

1. Find an equation for \(L_1\) in the form \(\mathbf{r} = \mathbf{a} + t\mathbf{b}\). [2]
2. Prove that \(L_1\) and \(L_2\) are skew. [5]

1. An octopus is able to catch any fish that swim within a distance of 2 m from the octopus's position.

A fish \(F\) swims from a point \(A\) to a point \(B\). The octopus is modelled as a fixed particle at the origin \(O\). Fish \(F\) is modelled as a particle moving in a straight line from \(A\) to \(B\). Relative to \(O\), the coordinates of \(A\) are \(( - 3,1 , - 7 )\) and the coordinates of \(B\) are \(( 9,4,11 )\), where the unit of distance is metres.

1. Use the model to determine whether or not the octopus is able to catch fish \(F\).
2. Criticise the model in relation to fish \(F\).
3. Criticise the model in relation to the octopus.

3 A particle is in equilibrium when acted on by the forces \(\left( \begin{array} { r } x \\ - 7 \\ z \end{array} \right) , \left( \begin{array} { r } 4 \\ y \\ - 5 \end{array} \right)\) and \(\left( \begin{array} { r } 5 \\ 4 \\ - 7 \end{array} \right)\), where the units are newtons.

1. Find the values of \(x , y\) and \(z\).
2. Calculate the magnitude of \(\left( \begin{array} { r } 5 \\ 4 \\ - 7 \end{array} \right)\).

4
2
- 6 \end{array} \right) + t \left( \begin{array} { r } - 8
1
- 2 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { r } - 2
a
- 2 \end{array} \right) + s \left( \begin{array} { r } - 9
2
- 5 \end{array} \right) ,$$ where a is \(a\) constant.

1. Calculate the acute angle between the lines.
2. Given that these two lines intersect, find \(a\) and the point of intersection. End of examination

10 \includegraphics[max width=\textwidth, alt={}, center]{0e4a249a-9e6a-49d4-996c-fe07b7730f59-16_318_1006_260_568} Relative to an origin \(O\), the position vectors of the points \(A , B , C\) and \(D\), shown in the diagram, are given by $$\overrightarrow { O A } = \left( \begin{array} { r } - 1 \\ 3 \\ - 4 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 2 \\ - 3 \\ 5 \end{array} \right) , \quad \overrightarrow { O C } = \left( \begin{array} { r } 4 \\ - 2 \\ 5 \end{array} \right) \quad \text { and } \quad \overrightarrow { O D } = \left( \begin{array} { r } 2 \\ 2 \\ - 1 \end{array} \right) .$$

1. Show that \(A B\) is perpendicular to \(B C\).
2. Show that \(A B C D\) is a trapezium.
3. Find the area of \(A B C D\), giving your answer correct to 2 decimal places.

1 The vectors \(\mathbf { a } , \mathbf { b } , \mathbf { c }\) and \(\mathbf { d }\) in \(\mathbb { R } ^ { 3 }\) are given by $$\mathbf { a } = \left( \begin{array} { r } 2 \\ - 1 \\ 1 \end{array} \right)$$

4. The line \(l _ { 1 }\) passes through the points \(P\) and \(Q\) with position vectors ( \(- \mathbf { i } - 8 \mathbf { j } + 3 \mathbf { k }\) ) and ( \(2 \mathbf { i } - 9 \mathbf { j } + \mathbf { k }\) ) respectively, relative to a fixed origin.

1. Find a vector equation for \(l _ { 1 }\). The line \(l _ { 2 }\) has the equation $$\mathbf { r } = ( 6 \mathbf { i } + a \mathbf { j } + b \mathbf { k } ) + t ( \mathbf { i } + 4 \mathbf { j } - \mathbf { k } )$$ and also passes through the point \(Q\).
2. Find the values of the constants \(a\) and \(b\).
3. Find, in degrees to 1 decimal place, the acute angle between lines \(l _ { 1 }\) and \(l _ { 2 }\).

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]

1. Find the coordinates of the point where the following three planes intersect. Give your answers in terms of \(a\). $$x - 2y - z = 6$$ $$3x + y + 5z = -4$$ $$-4x + 2y - 3z = a$$ [4]
2. Determine whether the intersection of the three planes could be on the \(z\)-axis. [2]

1. For the case where \(a = 2\), find the unit vector in the direction of \(\overrightarrow { P M }\).
2. For the case where angle \(A T P = \cos ^ { - 1 } \left( \frac { 2 } { 7 } \right)\), find the value of \(a\).

1
6 \end{array} \right) + \lambda \left( \begin{array} { r }

5. The vector equations of two straight lines are $$\begin{aligned} & \mathbf { r } = 5 \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } ) \quad \text { and } \\ & \mathbf { r } = 2 \mathbf { i } - 11 \mathbf { j } + a \mathbf { k } + \mu ( - 3 \mathbf { i } - 4 \mathbf { j } + 5 \mathbf { k } ) . \end{aligned}$$ Given that the two lines intersect, find

1. the coordinates of the point of intersection,
2. the value of the constant \(a\),
3. the acute angle between the two lines.

8. The line \(l _ { 1 }\) passes through the points \(A\) and \(B\) with position vectors \(( - 3 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k } )\) and ( \(7 \mathbf { i } - \mathbf { j } + 12 \mathbf { k }\) ) respectively, relative to a fixed origin.

1. Find a vector equation for \(l _ { 1 }\). The line \(l _ { 2 }\) has the equation $$\mathbf { r } = ( 5 \mathbf { j } - 7 \mathbf { k } ) + \mu ( \mathbf { i } - 2 \mathbf { j } + 7 \mathbf { k } )$$ The point \(C\) lies on \(l _ { 2 }\) and is such that \(A C\) is perpendicular to \(B C\).
2. Show that one possible position vector for \(C\) is \(( \mathbf { i } + 3 \mathbf { j } )\) and find the other. Assuming that \(C\) has position vector \(( \mathbf { i } + 3 \mathbf { j } )\),
3. find the area of triangle \(A B C\), giving your answer in the form \(k \sqrt { 5 }\).

10 Two intersecting straight lines have equations $$\frac { x - 5 } { 4 } = \frac { y - 11 } { 3 } = \frac { z - 7 } { - 5 } \quad \text { and } \quad \frac { x - 9 } { - 2 } = \frac { y - 4 } { 1 } = \frac { z + 4 } { 4 } .$$ Find the coordinates of their point of intersection.

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]