Vectors: Lines & Planes

1 Plane \(\Pi\) has equation \(3 x - y + 2 z = 33\). Line \(l\) has the following vector equation. $$l : \quad \mathbf { r } = \left( \begin{array} { l } 1
0
5 \end{array} \right) + \lambda \left( \begin{array} { l }

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 \(a x + b y + c z = 1\).

9 A line has Cartesian equations \(x - p = \frac { y + 2 } { q } = 3 - z\) and a plane has
equation r. \(\left[ \begin{array} { r } 1 \\ - 1 \\ - 2 \end{array} \right] = - 3\) 9

1. In the case where the plane fully contains the line, find the values of \(p\) and \(q\).
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   9
2. In the case where the line intersects the plane at a single point, find the range of values of \(p\) and \(q\).
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   9
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 \(z = 0\) 9
4. 1. Find the value of \(q\).
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5. (ii) Find the value of \(p\).

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 \(12 \mathbf { i } + 3 \mathbf { j } - 15 \mathbf { k }\) ．
（a）Find a vector for the line \(L _ { 1 }\) which passes through \(A\) and \(B\) ． The line \(L _ { 2 }\) has vector equation $$\mathbf { r } = - 4 \mathbf { i } + 12 \mathbf { k } + \mu ( \mathbf { i } - 3 \mathbf { k } )$$ （b）Show that \(L _ { 2 }\) passes through the origin \(O\) ．
（c）Show that \(L _ { 1 }\) and \(L _ { 2 }\) intersect at a point \(C\) and find the position vector of \(C\) ．
（d）Find the cosine of \(\angle O C A\) ．
（e）Hence，or otherwise，find the shortest distance from \(O\) to \(L _ { 1 }\) ．
（f）Show that \(| \overrightarrow { C O } | = | \overrightarrow { A B } |\) ．
（g）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 \(l\).

7 The plane \(\Pi _ { 1 }\) has equation \(r = - 4 \mathbf { j } - 3 \mathbf { k } + \lambda ( \mathbf { i } - \mathbf { j } + \mathbf { k } ) + \mu ( \mathbf { i } + \mathbf { j } - \mathbf { k } )\).

1. Obtain an equation of \(\Pi _ { 1 }\) in the form \(\mathrm { px } + \mathrm { qy } + \mathrm { rz } = \mathrm { d }\).
2. The plane \(\Pi _ { 2 }\) has equation \(\mathbf { r } . ( - 5 \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k } ) = 4\). Find a vector equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
   The line \(l\) passes through the point \(A\) with position vector \(a \mathbf { i } + a \mathbf { j } + ( a - 7 ) \mathbf { k }\) and is parallel to \(( 1 - b ) \mathbf { i } + b \mathbf { j } + b \mathbf { k }\), where \(a\) and \(b\) are positive constants.
3. Given that the perpendicular distance from \(A\) to \(\Pi _ { 1 }\) is \(\sqrt { 2 }\), find the value of \(a\).
4. Given that the obtuse angle between \(l\) and \(\Pi _ { 1 }\) is \(\frac { 3 } { 4 } \pi\), find the exact value of \(b\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

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

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.

2 A surface has equation \(z = 2 \left( x ^ { 3 } + y ^ { 3 } \right) + 3 \left( x ^ { 2 } + y ^ { 2 } \right) + 12 x y\).

1. For a point on the surface at which \(\frac { \partial z } { \partial x } = \frac { \partial z } { \partial y }\), show that either \(y = x\) or \(y = 1 - x\).
2. Show that there are exactly two stationary points on the surface, and find their coordinates.
3. The point \(\mathrm { P } \left( \frac { 1 } { 2 } , \frac { 1 } { 2 } , 5 \right)\) is on the surface, and \(\mathrm { Q } \left( \frac { 1 } { 2 } + h , \frac { 1 } { 2 } + h , 5 + w \right)\) is a point on the surface close to P . Find an approximate expression for \(h\) in terms of \(w\).
4. Find the four points on the surface at which the normal line is parallel to the vector \(24 \mathbf { i } + 24 \mathbf { j } - \mathbf { k }\).

9 The quadrilateral \(A B C D\) is a trapezium in which \(A B\) and \(D C\) are parallel. With respect to the origin \(O\), the position vectors of \(A , B\) and \(C\) are given by \(\overrightarrow { O A } = - \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } , \overrightarrow { O B } = \mathbf { i } + 3 \mathbf { j } + \mathbf { k }\) and \(\overrightarrow { O C } = 2 \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k }\).

1. Given that \(\overrightarrow { D C } = 3 \overrightarrow { A B }\), find the position vector of \(D\).
2. State a vector equation for the line through \(A\) and \(B\).
3. Find the distance between the parallel sides and hence find the area of the trapezium.

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.

9 The quadrilateral \(A B C D\) is a trapezium in which \(A B\) and \(D C\) are parallel. With respect to the origin \(O\), the position vectors of \(A , B\) and \(C\) are given by \(\overrightarrow { O A } = - \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } , \overrightarrow { O B } = \mathbf { i } + 3 \mathbf { j } + \mathbf { k }\) and \(\overrightarrow { O C } = 2 \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k }\).

1. Given that \(\overrightarrow { D C } = 3 \overrightarrow { A B }\), find the position vector of \(D\).
2. State a vector equation for the line through \(A\) and \(B\).
3. Find the distance between the parallel sides and hence find the area of the trapezium.

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.
   (a) Show that \(a = - \frac { 6 } { 13 }\).
   (b) Find a cartesian equation of the plane containing \(l _ { 1 }\) and \(l _ { 2 }\).
2. 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

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

1. Find the cosine of angle \(A B C\). The line \(L\) is the angle bisector of angle \(A B C\).
2. Show that an equation of \(L\) is \(\mathbf { r } = \mathbf { i } - \mathbf { k } + t ( \mathbf { i } + 2 \mathbf { j } - 7 \mathbf { k } )\).
3. Show that \(| \overrightarrow { A B } | = | \overrightarrow { A C } |\). The circle \(S\) lies inside triangle \(A B C\) and each side of the triangle is a tangent to \(S\).
4. Find the position vector of the centre of \(S\).
5. Find the radius of \(S\).

9 With respect to the origin \(O\), the point \(A\) has position vector given by \(\overrightarrow { O A } = \mathbf { i } + 5 \mathbf { j } + 6 \mathbf { k }\). The line \(l\) has vector equation \(\mathbf { r } = 4 \mathbf { i } + \mathbf { k } + \lambda ( - \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } )\).

1. Find in degrees the acute angle between the directions of \(O A\) and \(l\).
2. Find the position vector of the foot of the perpendicular from \(A\) to \(l\).
3. Hence find the position vector of the reflection of \(A\) in \(l\).

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 \(\sqrt { } 3\).

7. The plane \(\Pi\) has vector equation $$\mathbf { r } = 3 \mathbf { i } + \mathbf { k } + \lambda ( - 4 \mathbf { i } + \mathbf { j } ) + \mu ( 6 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } )$$

1. Find an equation of \(\Pi\) in the form \(\mathbf { r } \cdot \mathbf { n } = p\), where \(\mathbf { n }\) is a vector perpendicular to \(\Pi\) and \(p\) is a constant. The point \(P\) has coordinates \(( 6,13,5 )\). The line \(l\) passes through \(P\) and is perpendicular to \(\Pi\). The line \(l\) intersects \(\Pi\) at the point \(N\).
2. Show that the coordinates of \(N\) are \(( 3,1 , - 1 )\). The point \(R\) lies on \(\Pi\) and has coordinates \(( 1,0,2 )\).
3. Find the perpendicular distance from \(N\) to the line \(P R\). Give your answer to 3 significant figures.

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.

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

1. Find the angle between \(A B\) and \(A C\).
2. Find 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．
（a）Prove that \(\mathbf { p } . \mathbf { q } = | \mathbf { p } | ^ { 2 }\) ． \begin{figure}[h] \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
（b）the value of \(\lambda\) ，
（c）the position vector of \(R\) ，
（d）the area of \(K\) ． Another circle \(C _ { 1 }\) is drawn inside \(K\) so that the 4 sides of the kite are each tangents to \(C _ { 1 }\) ．
（e）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 \(C _ { 1 }\) ．
（f）Find that area of \(K _ { 1 }\) ．

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 \(O A B\).

6 A tetrahedron \(A B C D\) is such that \(A B\) is perpendicular to the base \(B C D\). The coordinates of the points \(A , C\) and \(D\) are \(( - 1 , - 7,2 ) , ( 5,0,3 )\) and \(( - 1,3,3 )\) respectively, and the equation of the plane \(B C D\) is \(x + 2 y - 2 z = - 1\).

1. Find, in either order, the coordinates of \(B\) and the length of \(A B\).
2. Find the acute angle between the planes \(A C D\) and \(B C D\).
3. (a) Verify, without using a calculator, that \(\theta = \frac { 1 } { 8 } \pi\) is a solution of the equation \(\sin 6 \theta = \sin 2 \theta\).
   (b) By sketching the graphs of \(y = \sin 6 \theta\) and \(y = \sin 2 \theta\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\), or otherwise, find the other solution of the equation \(\sin 6 \theta = \sin 2 \theta\) in the interval \(0 < \theta < \frac { 1 } { 2 } \pi\).
4. Use de Moivre's theorem to prove that $$\sin 6 \theta \equiv \sin 2 \theta \left( 16 \cos ^ { 4 } \theta - 16 \cos ^ { 2 } \theta + 3 \right) .$$
5. Hence show that one of the solutions obtained in part (i) satisfies \(\cos ^ { 2 } \theta = \frac { 1 } { 4 } ( 2 - \sqrt { 2 } )\), and justify which solution it is.

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 $$D A : D B = O A : O B$$

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

1. Use a scalar product to find angle \(B O A\). The point \(C\) is the mid-point of \(A B\). The point \(D\) is such that \(\overrightarrow { O D } = 2 \overrightarrow { O B }\).
2. Find \(\overrightarrow { D C }\).

7 Relative to an origin \(O\), the position vectors of the points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 1 \\ - 3 \\ 2 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } - 1 \\ 3 \\ 5 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } 3 \\ 1 \\ - 2 \end{array} \right)$$

1. Find \(\overrightarrow { A C }\).
2. The point \(M\) is the mid-point of \(A C\). Find the unit vector in the direction of \(\overrightarrow { O M }\).
3. Evaluate \(\overrightarrow { A B } \cdot \overrightarrow { A C }\) and hence find angle \(B A C\).

7
\includegraphics[max width=\textwidth, alt={}, center]{1cf37a58-8a7f-4dc8-9e35-2e8badf3eb83-3_636_1047_1153_550} The diagram shows a triangular prism with a horizontal rectangular base \(A D F C\), where \(C F = 12\) units and \(D F = 6\) units. The vertical ends \(A B C\) and \(D E F\) are isosceles triangles with \(A B = B C = 5\) units. The mid-points of \(B E\) and \(D F\) are \(M\) and \(N\) respectively. The origin \(O\) is at the mid-point of \(A C\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O C , O N\) and \(O B\) respectively.

1. Find the length of \(O B\).
2. Express each of the vectors \(\overrightarrow { M C }\) and \(\overrightarrow { M N }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
3. Evaluate \(\overrightarrow { M C } \cdot \overrightarrow { M N }\) and hence find angle \(C M N\), giving your answer correct to the nearest degree.

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.

10 The line \(l\) has equation \(\mathbf { r } = \mathbf { i } + \mathbf { j } + \mathbf { k } + \lambda ( a \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )\), where \(a\) is a constant. The plane \(p\) has equation \(x + 2 y + 2 z = 6\). Find the value or values of \(a\) in each of the following cases.

1. The line \(l\) is parallel to the plane \(p\).
2. The line \(l\) intersects the line passing through the points with position vectors \(3 \mathbf { i } + 2 \mathbf { j } + \mathbf { k }\) and \(\mathbf { i } + \mathbf { j } - \mathbf { k }\).
3. The acute angle between the line \(l\) and the plane \(p\) is \(\tan ^ { - 1 } 2\).

1 Find the acute angle between the lines with vector equations \(\mathbf { r } = \left( \begin{array} { c } 3 \\ 0 \\ - 2 \end{array} \right) + \lambda \left( \begin{array} { c } 1 \\ 2 \\ - 1 \end{array} \right)\) and \(\mathbf { r } = \left( \begin{array} { l } 1 \\ 5 \\ 3 \end{array} \right) + \mu \left( \begin{array} { c } 3 \\ 1 \\ - 2 \end{array} \right)\).

6
3 \end{array} \right) + \mu \left( \begin{array} { l } 1
0
2 \end{array} \right)$$ Find the acute angle between the lines. 4 A computer-controlled machine can be programmed to make cuts by entering the equation of the plane of the cut, and to drill boles by entering the equation of the line of the hole. A \(20 \mathrm {~cm} \times 30 \mathrm {~cm} \times 30 \mathrm {~cm}\) cuboid is to be cut and drilled. The cuboid is positioned relative to \(x - y ^ { 2 }\) and \(z\)-axes as shown in Fig. 8.1. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} First, a plane cut is made to remove the comer at \(E\). The cut goes through the points \(P , Q\) and \(R\), which are the midpoints of the sides \(\mathrm { ED } , \mathrm { EA }\) and EF respectively.

1. Write down the coordinates \(\boldsymbol { 0 } \mathbf { F } \mathrm { Q }\) and \(\mathrm { R } \left( \begin{array} { l } F \\ 1 \end{array} \right]\)
   Hence show that \(\mathrm { PQ } = { } _ { - }\): and \(\mathrm { PR } =\)
   (U) Show th,i tho \(, 0010,11\) is pc,pondio,la, to the pl'ute through \(P , Q\) rudd \(R\) Hence find the cartesian equation of this plane. A hole is then drilled perpendicular to triangle PQR , as shown in Fig. 82. The hole passes through the triangle at the point T which divides the line PS in the ratio 2 : I , where S is the midpoint of QR .
2. Write down the coordinates of S , and show that the point T has coordinates \(( - 5.16 \mathrm { i } , 25 )\).
3. Write down a vector equation of the line of the drill hole. Hence determine whether or not this line passes through C . 5 A tent has vertices ABCDEF with coordinates as shown in Fig. 7. Lengths are in metres. The Oxy plane is horizontal. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{253ddd65-d92b-46ce-bf17-b4f6e3d32ec0-4_555_1004_486_565} \captionsetup{labelformat=empty} \caption{Fig. 7}
   \end{figure}
4. Find the length of the ridge of the tent DE , and the angle this makes with the horizontal.
5. Show that the vector \(\mathbf { i } - 4 \mathbf { j } + 5 \mathbf { k }\) is normal to the plane through \(\mathrm { A } , \mathrm { D }\) and E . Hence find the equation of this plane. Given that B lies in this plane, find \(a\).
6. Verify that the equation of the plane BCD is \(x + z = 8\). Hence find the acute angle between the planes ABDE and BCD .

11 The Cartesian equation of the line \(L _ { 1 }\) is $$\frac { x + 1 } { 3 } = \frac { - y + 5 } { 2 } = \frac { 2 z + 5 } { 3 }$$ The Cartesian equation of the line \(L _ { 2 }\) is $$\frac { 2 x - 1 } { 2 } = \frac { y - 14 } { m } = \frac { z + 12 } { p }$$ The non-singular matrix \(\mathbf { N } = \left[ \begin{array} { c c c } - 0.5 & 1 & 2 \\ 1 & b & 4 \\ - 3 & - 2 & c \end{array} \right]\) maps the line \(L _ { 1 }\) onto the line \(L _ { 2 }\)
Calculate the values of the constants \(b , c , m\) and \(p\)
Fully justify your answers.

6 \end{array} \right) \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { r }

9 Relative to an origin \(O\), the position vectors of the points \(A , B , C\) and \(D\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 1 \\ 3 \\ - 1 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 3 \\ - 1 \\ 3 \end{array} \right) , \quad \overrightarrow { O C } = \left( \begin{array} { l } 4 \\ 2 \\ p \end{array} \right) \quad \text { and } \quad \overrightarrow { O D } = \left( \begin{array} { r } - 1 \\ 0 \\ q \end{array} \right) ,$$ where \(p\) and \(q\) are constants. Find

1. the unit vector in the direction of \(\overrightarrow { A B }\),
2. the value of \(p\) for which angle \(A O C = 90 ^ { \circ }\),
3. the values of \(q\) for which the length of \(\overrightarrow { A D }\) is 7 units.

10
\includegraphics[max width=\textwidth, alt={}, center]{56d4d40a-32f5-4f2d-938e-a24312cd42e7-4_552_629_842_758} The diagram shows the parallelogram \(O A B C\). Given that \(\overrightarrow { O A } = \mathbf { i } + 3 \mathbf { j } + 3 \mathbf { k }\) and \(\overrightarrow { O C } = 3 \mathbf { i } - \mathbf { j } + \mathbf { k }\), find

1. the unit vector in the direction of \(\overrightarrow { O B }\),
2. the acute angle between the diagonals of the parallelogram,
3. the perimeter of the parallelogram, correct to 1 decimal place. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }

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 \(b\).

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 \(l\).

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 \(q\).

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

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.

11 The line \(l _ { 1 }\) passes through the points \(A ( 6,2,7 )\) and \(B ( 4 , - 3,7 )\) 11

1. Find a Cartesian equation of \(l _ { 1 }\)
   11
2. The line \(l _ { 2 }\) has vector equation \(\mathbf { r } = \left[ \begin{array} { l } 8 \\ 9 \\ c \end{array} \right] + \mu \left[ \begin{array} { l } 1 \\ 1 \\ 2 \end{array} \right]\) where \(c\) is a constant.
   11
3. 1. Explain how you know that the lines \(l _ { 1 }\) and \(l _ { 2 }\) are not perpendicular.
      11
4. (ii) The lines \(l _ { 1 }\) and \(l _ { 2 }\) both lie in the same plane. Find the value of \(c\)

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 \(U V W\).