Vectors: Lines & Planes

3
1 \end{array} \right) + \lambda \left( \begin{array} { r } - 2
4
- 2 \end{array} \right)
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { l }

9 With respect to the origin \(O\), the position vectors of the points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \left( \begin{array} { l } 0 \\ 5 \\ 2 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { l } 1 \\ 0 \\ 1 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } 4 \\ - 3 \\ - 2 \end{array} \right)$$ The midpoint of \(A C\) is \(M\) and the point \(N\) lies on \(B C\), between \(B\) and \(C\), and is such that \(B N = 2 N C\).

1. Find the position vectors of \(M\) and \(N\).
2. Find a vector equation for the line through \(M\) and \(N\).
3. Find the position vector of the point \(Q\) where the line through \(M\) and \(N\) intersects the line through \(A\) and \(B\).

5.The point \(A\) has position vector \(7 \mathbf { i } + 2 \mathbf { j } - 7 \mathbf { k }\) and the point \(B\) has position vector \(12 \mathbf { i } + 3 \mathbf { j } - 15 \mathbf { k }\) .

1. Find a vector for the line \(L _ { 1 }\) which passes through \(A\) and \(B\) . The line \(L _ { 2 }\) has vector equation $$\mathbf { r } = - 4 \mathbf { i } + 12 \mathbf { k } + \mu ( \mathbf { i } - 3 \mathbf { k } )$$
2. Show that \(L _ { 2 }\) passes through the origin \(O\) .
3. Show that \(L _ { 1 }\) and \(L _ { 2 }\) intersect at a point \(C\) and find the position vector of \(C\) .
4. Find the cosine of \(\angle O C A\) .
5. Hence,or otherwise,find the shortest distance from \(O\) to \(L _ { 1 }\) .
6. Show that \(| \overrightarrow { C O } | = | \overrightarrow { A B } |\) .
7. Find a vector equation for the line which bisects \(\angle O C A\) . \includegraphics[max width=\textwidth, alt={}, center]{f9d3e02c-cef2-435b-9cda-76c43fcac575-4_922_1054_279_586} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = x \left( 12 - x ^ { 2 } \right) .$$ The curve cuts the \(x\)-axis at the points \(P , O\) and \(R\), and \(Q\) is the maximum point.

7 Two planes have equations \(2 x - y - 3 z = 7\) and \(x + 2 y + 2 z = 0\).

1. Find the acute angle between the planes.
2. Find a vector equation for their line of intersection.

9 Two planes have equations \(x + 2 y - 2 z = 2\) and \(2 x - 3 y + 6 z = 3\). The planes intersect in the straight line \(l\).

1. Calculate the acute angle between the two planes.
2. Find a vector equation for the line \(l\).

A surface \(S\) has equation \(g(x, y, z) = 0\), where \(g(x, y, z) = (y - 2x)(y + z)^2 - 18\).

1. Show that \(\frac{\partial g}{\partial y} = (y + z)(-4x + 3y + z)\). [2]
2. Show that \(\frac{\partial g}{\partial x} + 2\frac{\partial g}{\partial y} - 2\frac{\partial g}{\partial z} = 0\). [4]
3. Hence identify a vector which lies in the tangent plane of every point on \(S\), explaining your reasoning. [3]
4. Find the cartesian equation of the tangent plane to the surface \(S\) at the point P\((1, 4, -7)\). [3]

The tangent plane to the surface \(S\) at the point Q\((0, 2, 1)\) has equation \(6x - 7y - 4z = -18\).

1. Find a vector equation for the line of intersection of the tangent planes at P and Q. [4]

6 The straight line \(l\) passes through the points with coordinates \(( - 5,3,6 )\) and \(( 5,8,1 )\). The plane \(p\) has equation \(2 x - y + 4 z = 9\).

1. Find the coordinates of the point of intersection of \(l\) and \(p\).
2. Find the acute angle between \(l\) and \(p\).

8 The straight line \(l\) passes through the points \(A\) and \(B\) whose position vectors are \(\mathbf { i } + \mathbf { k }\) and \(4 \mathbf { i } - \mathbf { j } + 3 \mathbf { k }\) respectively. The plane \(p\) has equation \(x + 3 y - 2 z = 3\).

1. Given that \(l\) intersects \(p\), find the position vector of the point of intersection.
2. Find the equation of the plane which contains \(l\) and is perpendicular to \(p\), giving your answer in the form \(a x + b y + c z = 1\).

11 The plane \(\Pi _ { 1 }\) has equation $$\mathbf { r } = \mathbf { i } + 2 \mathbf { j } + \mathbf { k } + \theta ( 2 \mathbf { j } - \mathbf { k } ) + \phi ( 3 \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k } )$$ Find a vector normal to \(\Pi _ { 1 }\) and hence show that the equation of \(\Pi _ { 1 }\) can be written as \(2 x + 3 y + 6 z = 14\). The line \(l\) has equation $$\mathbf { r } = 3 \mathbf { i } + 8 \mathbf { j } + 2 \mathbf { k } + t ( 4 \mathbf { i } + 6 \mathbf { j } + 5 \mathbf { k } )$$ The point on \(l\) where \(t = \lambda\) is denoted by \(P\). Find the set of values of \(\lambda\) for which the perpendicular distance of \(P\) from \(\Pi _ { 1 }\) is not greater than 4 . The plane \(\Pi _ { 2 }\) contains \(l\) and the point with position vector \(\mathbf { i } + 2 \mathbf { j } + \mathbf { k }\). Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).

Find the cartesian equation of the plane which contains the three points \((1, 0, -1)\), \((2, 2, 1)\) and \((1, 1, 2)\). [5]

3 Points \(A\) and \(B\) have coordinates \(( - 1,2,5 )\) and \(( 2 , - 2,11 )\) respectively. The plane \(p\) passes through \(B\) and is perpendicular to \(A B\).

1. Find an equation of \(p\), giving your answer in the form \(a x + b y + c z = d\).
2. Find the acute angle between \(p\) and the \(y\)-axis.

7 The points \(A , B , C\) have position vectors $$- 2 \mathbf { i } + 2 \mathbf { j } - \mathbf { k } , \quad - 2 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } , \quad - 2 \mathbf { j } + \mathbf { k } ,$$ respectively, relative to the origin \(O\).

1. Find the equation of the plane \(A B C\), giving your answer in the form \(a x + b y + c z = d\).
2. Find the acute angle between the planes \(O B C\) and \(A B C\).
   The point \(D\) has position vector \(t \mathbf { i } - \mathbf { j }\).
3. Given that the shortest distance between the lines \(A B\) and \(C D\) is \(\sqrt { \mathbf { 1 0 } }\), find the value of \(t\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

The line \(L_1\) has equation $$\frac{x-2}{3} = \frac{y+4}{8} = \frac{4z-5}{5}$$ The line \(L_2\) has equation $$\left(\mathbf{r} - \begin{bmatrix} -2 \\ 0 \\ 3 \end{bmatrix}\right) \times \begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix} = \mathbf{0}$$ Find the shortest distance between the two lines, giving your answer to three significant figures. [8 marks]

10 With respect to the origin \(O\), the points \(A , B , C , D\) have position vectors given by $$\overrightarrow { O A } = 4 \mathbf { i } + \mathbf { k } , \quad \overrightarrow { O B } = 5 \mathbf { i } - 2 \mathbf { j } - 2 \mathbf { k } , \quad \overrightarrow { O C } = \mathbf { i } + \mathbf { j } , \quad \overrightarrow { O D } = - \mathbf { i } - 4 \mathbf { k }$$

1. Calculate the acute angle between the lines \(A B\) and \(C D\).
2. Prove that the lines \(A B\) and \(C D\) intersect.
3. The point \(P\) has position vector \(\mathbf { i } + 5 \mathbf { j } + 6 \mathbf { k }\). Show that the perpendicular distance from \(P\) to the line \(A B\) is equal to \(\sqrt { } 3\).

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]

1. Find the value of \(a\).
2. When \(a\) has this value, find the equation of the plane containing \(l\) and \(m\).

10 The points \(A\) and \(B\) have position vectors given by \(\overrightarrow { O A } = 2 \mathbf { i } - \mathbf { j } + 3 \mathbf { k }\) and \(\overrightarrow { O B } = \mathbf { i } + \mathbf { j } + 5 \mathbf { k }\). The line \(l\) has equation \(\mathbf { r } = \mathbf { i } + \mathbf { j } + 2 \mathbf { k } + \mu ( 3 \mathbf { i } + \mathbf { j } - \mathbf { k } )\).

1. Show that \(l\) does not intersect the line passing through \(A\) and \(B\).
2. Find the equation of the plane containing the line \(l\) and the point \(A\). Give your answer in the form \(a x + b y + c z = d\).

10 The line \(l _ { 1 }\) is parallel to the vector \(a \mathbf { i } - \mathbf { j } + \mathbf { k }\), where \(a\) is a constant, and passes through the point whose position vector is \(9 \mathbf { j } + 2 \mathbf { k }\). The line \(l _ { 2 }\) is parallel to the vector \(- a \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k }\) and passes through the point whose position vector is \(- 6 \mathbf { i } - 5 \mathbf { j } + 10 \mathbf { k }\).

1. It is given that \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
   1. Show that \(a = - \frac { 6 } { 13 }\).
   2. Find a cartesian equation of the plane containing \(l _ { 1 }\) and \(l _ { 2 }\).
   3. Given instead that the perpendicular distance between \(l _ { 1 }\) and \(l _ { 2 }\) is \(3 \sqrt { } ( 30 )\), find the value of \(a\).

10 The line \(l\) has vector equation \(\mathbf { r } = \mathbf { i } + 2 \mathbf { j } + \mathbf { k } + \lambda ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } )\).

1. Find the position vectors of the two points on the line whose distance from the origin is \(\sqrt { } ( 10 )\).
2. The plane \(p\) has equation \(a x + y + z = 5\), where \(a\) is a constant. The acute angle between the line \(l\) and the plane \(p\) is equal to \(\sin ^ { - 1 } \left( \frac { 2 } { 3 } \right)\). Find the possible values of \(a\).

8. With respect to a fixed origin \(O\) the points \(A\) and \(B\) have position vectors $$\left( \begin{array} { l } 6 \\ 6 \\ 2 \end{array} \right) \text { and } \left( \begin{array} { l } 6 \\ 0 \\ 7 \end{array} \right)$$ respectively. The line \(l _ { 1 }\) passes through the points \(A\) and \(B\).

1. Write down an equation for \(l _ { 1 }\) Give your answer in the form \(\mathbf { r } = \mathbf { p } + \lambda \mathbf { q }\), where \(\lambda\) is a scalar parameter. The line \(l _ { 2 }\) has equation $$\mathbf { r } = \left( \begin{array} { l } 3 \\ 1 \\ 4 \end{array} \right) + \mu \left( \begin{array} { l } 1 \\ 5 \\ 9 \end{array} \right)$$ where \(\mu\) is a scalar parameter.
2. Show that \(l _ { 1 }\) and \(l _ { 2 }\) do not meet. The point \(C\) is on \(l _ { 2 }\) where \(\mu = - 1\)
3. Find the acute angle between \(A C\) and \(l _ { 2 }\) Give your answer in degrees to one decimal place. \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \section*{Question 8 continued} \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \section*{Question 8 continued} \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \section*{Question 8 continued}
   1. (a) Find the derivative with respect to \(y\) of
   $$\frac { 1 } { ( 1 + 2 \ln y ) ^ { 2 } }$$
4. Hence find a general solution to the differential equation $$3 \operatorname { cosec } ( 2 x ) \frac { \mathrm { d } y } { \mathrm {~d} x } = y ( 1 + 2 \ln y ) ^ { 3 } \quad y > 0 \quad - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$$
5. Show that the particular solution of this differential equation for which \(y = 1\) at \(x = \frac { \pi } { 6 }\) is given by $$y = \mathrm { e } ^ { A \sec x - \frac { 1 } { 2 } }$$ where \(A\) is an irrational number to be found. \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \section*{Question 9 continued} \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \section*{Question 9 continued} \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \section*{Question 9 continued} \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \includegraphics[max width=\textwidth, alt={}, center]{fe07afad-9cfc-48c0-84f1-5717f81977d4-32_2649_1894_109_173}

1. The line \(L _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { c } - 13 \\ 7 \\ - 1 \end{array} \right) + t \left( \begin{array} { c } 6 \\ - 2 \\ 3 \end{array} \right)\). The line \(L _ { 2 }\) passes through the point \(A\) with position vector \(\left( \begin{array} { c } 1 \\ p \\ 10 \end{array} \right)\) and is parallel to \(\left( \begin{array} { c } - 2 \\ 11 \\ - 5 \end{array} \right)\), where \(p\) is a constant. The lines \(L _ { 1 }\) and \(L _ { 2 }\) intersect at the point \(B\).
   1. Find
      1. the value of \(p\),
      2. the position vector of \(B\).
   The point \(C\) lies on \(L _ { 1 }\) and angle \(A C B\) is \(90 ^ { \circ }\)
2. Find the position vector of \(C\). The point \(D\) also lies on \(L _ { 1 }\) and triangle \(A B D\) is isosceles with \(A B = A D\).
3. Find the area of triangle \(A B D\).

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]

7 A straight pipeline AB passes through a mountain. With respect to axes \(\mathrm { O } x y z\), with \(\mathrm { O } x\) due East, \(\mathrm { O } y\) due North and \(\mathrm { O } z\) vertically upwards, A has coordinates \(( - 200,100,0 )\) and B has coordinates \(( 100,200,100 )\), where units are metres.

1. Verify that \(\overrightarrow { \mathrm { AB } } = \left( \begin{array} { l } 300 \\ 100 \\ 100 \end{array} \right)\) and find the length of the pipeline.
2. Write down a vector equation of the line AB , and calculate the angle it makes with the vertical. A thin flat layer of hard rock runs through the mountain. The equation of the plane containing this layer is \(x + 2 y + 3 z = 320\).
3. Find the coordinates of the point where the pipeline meets the layer of rock.
4. By calculating the angle between the line AB and the normal to the plane of the layer, find the angle at which the pipeline cuts through the layer.

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]

5 Two planes, \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), have equations \(3 x + 2 y + z = 4\) and \(2 x + y + z = 3\) respectively.

1. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\). The line \(L\) has equation \(x = 1 - y = 2 - z\).
2. Show that \(L\) lies in both planes.

9 \includegraphics[max width=\textwidth, alt={}, center]{8580dddb-cc72-4745-9e0f-1ac641c6506d-3_693_537_1206_804} The diagram shows a set of rectangular axes \(O x , O y\) and \(O z\), and three points \(A , B\) and \(C\) with position vectors \(\overrightarrow { O A } = \left( \begin{array} { l } 2 \\ 0 \\ 0 \end{array} \right) , \overrightarrow { O B } = \left( \begin{array} { l } 1 \\ 2 \\ 0 \end{array} \right)\) and \(\overrightarrow { O C } = \left( \begin{array} { l } 1 \\ 1 \\ 2 \end{array} \right)\).

1. Find the equation of the plane \(A B C\), giving your answer in the form \(a x + b y + c z = d\).
2. Calculate the acute angle between the planes \(A B C\) and \(O A B\).

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 \(\cos ^ { - 1 } \left( \frac { 2 } { 3 } \right)\).

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

2 Write down normal vectors to the planes \(2 x + 3 y + 4 z = 10\) and \(x - 2 y + z = 5\).
Hence show that these planes are perpendicular to each other.

10 The coordinates of the points \(A\) and \(B\) are ( \(3 , - 2 , - 1\) ) and ( \(13,10,9\) ) respectively.

• The plane \(\Pi _ { A }\) contains \(A\) and the plane \(\Pi _ { B }\) contains \(B\).
• The planes \(\Pi _ { A }\) and \(\Pi _ { B }\) are parallel.
• The \(x\) and \(y\) components of any normal to plane \(\Pi _ { A }\) are equal.
• The shortest distance between \(\Pi _ { A }\) and \(\Pi _ { B }\) is 2 .

There are two possible solution planes for \(\Pi _ { A }\) which satisfy the above conditions.
Determine the acute angle between these two possible solution planes.

7 The position vectors of points \(A , B\) and \(C\) relative to an origin \(O\) are given by $$\overrightarrow { O A } = \left( \begin{array} { l } 2 \\ 1 \\ 3 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 6 \\ - 1 \\ 7 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { l } 2 \\ 4 \\ 7 \end{array} \right)$$

1. Show that angle \(B A C = \cos ^ { - 1 } \left( \frac { 1 } { 3 } \right)\).
2. Use the result in part (i) to find the exact value of the area of triangle \(A B C\).

6 Relative to the origin \(O\), the points \(A , B\) and \(C\) have position vectors given by $$\overrightarrow { O A } = \left( \begin{array} { l } 2 \\ 1 \\ 3 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { l } 4 \\ 3 \\ 2 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } 3 \\ - 2 \\ - 4 \end{array} \right) .$$ The quadrilateral \(A B C D\) is a parallelogram.

1. Find the position vector of \(D\).
2. The angle between \(B A\) and \(B C\) is \(\theta\). Find the exact value of \(\cos \theta\).
3. Hence find the area of \(A B C D\), giving your answer in the form \(p \sqrt { q }\), where \(p\) and \(q\) are integers.

7.The points \(O , P\) and \(Q\) lie on a circle \(C\) with diameter \(O Q\) .The position vectors of \(P\) and \(Q\) , relative to \(O\) ,are \(\mathbf { p }\) and \(\mathbf { q }\) respectively.

1. Prove that \(\mathbf { p } . \mathbf { q } = | \mathbf { p } | ^ { 2 }\) . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f2290882-b9a4-43ec-a38f-c44d46477242-6_615_714_412_689} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} The point \(R\) also lies on \(C\) and \(O P Q R\) is a kite \(K\) as shown in Figure 3.The point \(S\) has position vector,relative to \(O\) ,of \(\lambda \mathbf { q }\) ,where \(\lambda\) is a constant.Given that \(\mathbf { p } = \mathbf { i } + 2 \mathbf { j } - \mathbf { k } , \mathbf { q } = 2 \mathbf { i } + \mathbf { j } - 2 \mathbf { k }\) and that \(O Q\) is perpendicular to \(P S\) ,find
2. the value of \(\lambda\) ,
3. the position vector of \(R\) ,
4. the area of \(K\) . Another circle \(C _ { 1 }\) is drawn inside \(K\) so that the 4 sides of the kite are each tangents to \(C _ { 1 }\) .
5. Find the radius of \(C _ { 1 }\) giving your answer in the form \(( \sqrt { } 2 - 1 ) \sqrt { } n\) ,where \(n\) is an integer. A second kite \(K _ { 1 }\) is similar to \(K\) and is drawn inside \(C _ { 1 }\) .
6. Find that area of \(K _ { 1 }\) .

7 Given that \(\mathbf { a } = \left( \begin{array} { r } 2 \\ - 2 \\ 1 \end{array} \right) , \mathbf { b } = \left( \begin{array} { l } 2 \\ 6 \\ 3 \end{array} \right)\) and \(\mathbf { c } = \left( \begin{array} { c } p \\ p \\ p + 1 \end{array} \right)\), find

1. the angle between the directions of \(\mathbf { a }\) and \(\mathbf { b }\),
2. the value of \(p\) for which \(\mathbf { b }\) and \(\mathbf { c }\) are perpendicular.

7 Given that \(\mathbf { a } = \left( \begin{array} { r } 2 \\ - 2 \\ 1 \end{array} \right) , \mathbf { b } = \left( \begin{array} { l } 2 \\ 6 \\ 3 \end{array} \right)\) and \(\mathbf { c } = \left( \begin{array} { c } p \\ p \\ p + 1 \end{array} \right)\), find

1. the angle between the directions of \(\mathbf { a }\) and \(\mathbf { b }\),
2. the value of \(p\) for which \(\mathbf { b }\) and \(\mathbf { c }\) are perpendicular.

1. Points \(A\) and \(B\) have position vectors \(\mathbf { a }\) and \(\mathbf { b }\), respectively, relative to an origin \(O\), and are such that \(O A B\) is a triangle with \(O A = a\) and \(O B = b\).

The point \(C\), with position vector \(\mathbf { c }\), lies on the line through \(O\) that bisects the angle \(A O B\).

1. Prove that the vector \(b \mathbf { a } - a \mathbf { b }\) is perpendicular to \(\mathbf { c }\). The point \(D\), with position vector \(\mathbf { d }\), lies on the line \(A B\) between \(A\) and \(B\).
2. Explain why \(\mathbf { d }\) can be expressed in the form \(\mathbf { d } = ( 1 - \lambda ) \mathbf { a } + \lambda \mathbf { b }\) for some scalar \(\lambda\) with \(0 < \lambda < 1\)
3. Given that \(D\) is also on the line \(O C\), find an expression for \(\lambda\) in terms of \(a\) and \(b\) only and hence show that $$D A : D B = O A : O B$$

2 In this question you must show detailed reasoning.
Find the angle between the vector \(3 i + 2 j + \mathbf { k }\) and the plane \(- x + 3 y + 2 z = 8\).

4 Determine the acute angle between the line \(\mathbf { r } = \left( \begin{array} { c } - \sqrt { 3 } \\ 1 \\ 3 \end{array} \right) + \lambda \left( \begin{array} { c } 1 \\ 2 \sqrt { 3 } \\ - \sqrt { 3 } \end{array} \right)\) and the \(y\)-axis.

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 line \(l\) passes through the point \(P(2, 1, 3)\) and is perpendicular to the plane \(\Pi\) whose vector equation is $$\mathbf{r} \cdot (\mathbf{i} - 2\mathbf{j} - \mathbf{k}) = 3$$ Find

1. a vector equation of the line \(l\), [2]
2. the position vector of the point where \(l\) meets \(\Pi\). [4]
3. Hence find the perpendicular distance of \(P\) from \(\Pi\). [2]

7 The straight line \(l\) has equation \(\mathbf { r } = 4 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } + \lambda ( 2 \mathbf { i } - 3 \mathbf { j } + 6 \mathbf { k } )\). The plane \(p\) passes through the point \(( 4 , - 1,2 )\) and is perpendicular to \(l\).

1. Find the equation of \(p\), giving your answer in the form \(a x + b y + c z = d\).
2. Find the perpendicular distance from the origin to \(p\).
3. A second plane \(q\) is parallel to \(p\) and the perpendicular distance between \(p\) and \(q\) is 14 units. Find the possible equations of \(q\).

7 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have vector equations $$\mathbf { r } = a \mathbf { i } + 9 \mathbf { j } + 13 \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = - 3 \mathbf { i } + 7 \mathbf { j } - 2 \mathbf { k } + \mu ( - \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k } )$$ respectively. It is given that \(l _ { 1 }\) and \(l _ { 2 }\) intersect.

1. Find the value of the constant \(a\).
   The point \(P\) has position vector \(3 \mathbf { i } + \mathbf { j } + 6 \mathbf { k }\).
2. Find the perpendicular distance from \(P\) to the plane containing \(l _ { 1 }\) and \(l _ { 2 }\).
3. Find the perpendicular distance from \(P\) to \(l _ { 2 }\).

1. The plane \(\Pi\) has equation

$$3 x + 4 y - z = 17$$ The line \(l _ { 1 }\) is perpendicular to \(\Pi\) and passes through the point \(P ( - 4 , - 5,3 )\) The line \(l _ { 1 }\) intersects \(\Pi\) at the point \(Q\)

1. Determine the coordinates of \(Q\) Given that the point \(R ( - 1,6,4 )\) lies on \(\Pi\)
2. determine a Cartesian equation of the plane containing \(P Q R\) The line \(l _ { 2 }\) passes through \(P\) and \(R\) The line \(l _ { 3 }\) is the reflection of \(l _ { 2 }\) in \(\Pi\)
3. Determine a vector equation for \(l _ { 3 }\)

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

• the point \(A\) has position vector \(4 \mathbf { i } + 8 \mathbf { j } + \mathbf { k }\)
• the point \(B\) has position vector \(5 \mathbf { i } + 6 \mathbf { j } + 3 \mathbf { k }\)
• the point \(P\) has position vector \(2 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }\)

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

1. Find a vector equation for \(l\). The point \(C\) lies on \(l\) so that \(P C\) is perpendicular to \(l\).
2. Find the coordinates of \(C\). The point \(P ^ { \prime }\) is the reflection of \(P\) in the line \(l\).
3. Find the coordinates of \(P ^ { \prime }\)
4. Hence find \(\left| \overrightarrow { P P ^ { \prime } } \right|\), giving your answer as a simplified surd.

5 The line \(l _ { 1 }\) has equation \(\frac { x + 5 } { 1 } = \frac { y + 4 } { - 3 } = \frac { z - 3 } { 5 }\) The plane \(\Pi _ { 1 }\) has equation \(2 x + 3 y - 2 z = 6\)

1. Find the point of intersection of \(l _ { 1 }\) and \(\Pi _ { 1 }\) The line \(l _ { 2 }\) is the reflection of the line \(l _ { 1 }\) in the plane \(\Pi _ { 1 }\)
2. Show that a vector equation for the line \(l _ { 2 }\) is $$\mathbf { r } = \left( \begin{array} { r } - 7 \\ 2 \\ - 7 \end{array} \right) + \mu \left( \begin{array} { c } 10 \\ 6 \\ 2 \end{array} \right)$$ where \(\mu\) is a scalar parameter. The plane \(\Pi _ { 2 }\) contains the line \(l _ { 1 }\) and the line \(l _ { 2 }\)
3. Determine a vector equation for the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) The plane \(\Pi _ { 3 }\) has equation r. \(\left( \begin{array} { l } 1 \\ 1 \\ a \end{array} \right) = b\) where \(a\) and \(b\) are constants.
   Given that the planes \(\Pi _ { 1 } , \Pi _ { 2 }\) and \(\Pi _ { 3 }\) form a sheaf,
4. determine the value of \(a\) and the value of \(b\).

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]

3 Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are given by $$\overrightarrow { O A } = 2 \mathbf { i } - 5 \mathbf { j } - 2 \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = 4 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k }$$ The point \(C\) is such that \(\overrightarrow { A B } = \overrightarrow { B C }\). Find the unit vector in the direction of \(\overrightarrow { O C }\).

8 The points \(A\) and \(B\) have position vectors \(\mathbf { i } + 7 \mathbf { j } + 2 \mathbf { k }\) and \(- 5 \mathbf { i } + 5 \mathbf { j } + 6 \mathbf { k }\) respectively, relative to an origin \(O\).

1. Use a scalar product to calculate angle \(A O B\), giving your answer in radians correct to 3 significant figures.
2. The point \(C\) is such that \(\overrightarrow { A B } = 2 \overrightarrow { B C }\). Find the unit vector in the direction of \(\overrightarrow { O C }\).

15 The equations of three planes are $$\begin{aligned} - 4 x + k y + 7 z & = 4 \\ x - 2 y + 5 z & = 1 \\ 2 x + 3 y + z & = 2 \end{aligned}$$ Given that the planes form a sheaf, determine the values of \(k\) and \(l\).

1

1. Find a vector which is perpendicular to both \(\left( \begin{array} { r } 1 \\ 3 \\ - 2 \end{array} \right)\) and \(\left( \begin{array} { r } - 3 \\ - 6 \\ 4 \end{array} \right)\).
2. The cartesian equation of a line is \(\frac { x } { 2 } = y - 3 = 2 z + 4\). Express the equation of this line in vector form.

1. The plane \(\Pi _ { 1 }\) has vector equation

$$\mathbf { r } = \left( \begin{array} { r } 1 \\ - 1 \\ 2 \end{array} \right) + s \left( \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right) + t \left( \begin{array} { r } 1 \\ 2 \\ - 2 \end{array} \right) ,$$ where \(s\) and \(t\) are real parameters. The plane \(\Pi _ { 1 }\) is transformed to the plane \(\Pi _ { 2 }\) by the transformation represented by the matrix \(\mathbf { T }\), where $$\mathbf { T } = \left( \begin{array} { r r r } 2 & 0 & 3 \\ 0 & 2 & - 1 \\ 0 & 1 & 2 \end{array} \right)$$ Find an equation of the plane \(\Pi _ { 2 }\) in the form r.n=p

7 With respect to the origin \(O\), the position vectors of the points \(U , V\) and \(W\) are \(\mathbf { u } , \mathbf { v }\) and \(\mathbf { w }\) respectively. The mid-points of the sides \(V W , W U\) and \(U V\) of the triangle \(U V W\) are \(M , N\) and \(P\) respectively.

1. Show that \(\overrightarrow { U M } = \frac { 1 } { 2 } ( \mathbf { v } + \mathbf { w } - 2 \mathbf { u } )\).
2. Verify that the point \(G\) with position vector \(\frac { 1 } { 3 } ( \mathbf { u } + \mathbf { v } + \mathbf { w } )\) lies on \(U M\), and deduce that the lines \(U M , V N\) and \(W P\) intersect at \(G\).
3. Write down, in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\), an equation of the line through \(G\) which is perpendicular to the plane \(U V W\). (It is not necessary to simplify the expression for \(\mathbf { b }\).)
4. It is now given that \(\mathbf { u } = \left( \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right) , \mathbf { v } = \left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)\) and \(\mathbf { w } = \left( \begin{array} { l } 0 \\ 0 \\ 1 \end{array} \right)\). Find the perpendicular distance from \(O\) to the plane \(U V W\).

4 The line \(L\) has vector equation $$\mathbf { r } = \left[ \begin{array} { c } 4 \\ - 7 \\ 0 \end{array} \right] + \lambda \left[ \begin{array} { c } - 9 \\ 1 \\ 3 \end{array} \right]$$ Give the equation of \(L\) in Cartesian form.
Tick ( ✓ ) one box. \(\frac { x + 4 } { - 9 } = \frac { y - 7 } { 1 } = \frac { z } { 3 }\) \includegraphics[max width=\textwidth, alt={}, center]{47b12ae4-ca3f-472c-9d15-2ef17a2a4d87-03_108_109_1398_993} \(\frac { x - 4 } { - 9 } = \frac { y + 7 } { 1 } = \frac { z } { 3 }\) \includegraphics[max width=\textwidth, alt={}, center]{47b12ae4-ca3f-472c-9d15-2ef17a2a4d87-03_108_111_1567_991} \(\frac { x + 9 } { 4 } = \frac { y - 1 } { - 7 } , z = 3\) □ \(\frac { x - 9 } { 4 } = \frac { y + 1 } { - 7 } , z = 3\) □