1.10d Vector operations: addition and scalar multiplication

1 A glass ornament OABCDEFG is a truncated pyramid on a rectangular base (see Fig. 7). All dimensions are in centimetres. \begin{figure}[h] \end{figure}

1. Write down the vectors \(\overrightarrow { \mathrm { CD } }\) and \(\overrightarrow { \mathrm { CB } }\).
2. Find the length of the edge CD.
3. Show that the vector \(4 \mathbf { i } + \mathbf { k }\) is perpendicular to the vectors \(\overrightarrow { \mathrm { CD } }\) and \(\overrightarrow { \mathrm { CB } }\). Hence find the cartesian equation of the plane BCDE.
4. Write down vector equations for the lines OG and AF . Show that they meet at the point P with coordinates (5, 10, 40). You may assume that the lines CD and BE also meet at the point P .
   The volume of a pyramid is \(\frac { 1 } { 3 } \times\) area of base × height.
5. Find the volumes of the pyramids POABC and PDEFG . Hence find the volume of the ornament.

3 Fig. 6 shows a lean-to greenhouse ABCDHEFG . With respect to coordinate axes Oxyz , the coordinates of the vertices are as shown. All distances are in metres. Ground level is the plane \(z = 0\). \begin{figure}[h] \end{figure}

1. Verify that the equation of the plane through \(\mathrm { A } , \mathrm { B }\) and E is \(x + 6 y + 12 = 0\). Hence, given that F lies in this plane, show that \(a = - 2 \frac { 1 } { 3 }\).
2. (A) Show that the vector \(\left( \begin{array} { r } 1 \\ - 6 \\ 0 \end{array} \right)\) is normal to the plane DHC.
   (B) Hence find the cartesian equation of this plane.
   (C) Given that G lies in the plane DHC , find \(b\) and the length FG .
3. Find the angle EFB . A straight wire joins point H to a point P which is half way between E and \(\mathrm { F } . \mathrm { Q }\) is a point two-thirds of the way along this wire, so that \(\mathrm { HQ } = 2 \mathrm { QP }\).
4. Find the height of Q above the ground.

1 Fig. 6 shows a lean-to greenhouse ABCDHEFG . With respect to coordinate axes \(\mathrm { O } x y z\), the coordinates of the vertices are as shown. All distances are in metres. Ground level is the plane \(z = 0\). \begin{figure}[h] \end{figure}

1. Verify that the equation of the plane through \(\mathrm { A } , \mathrm { B }\) and E is \(x + 6 y + 12 = 0\). Hence, given that F lies in this plane, show that \(a = - 2 \frac { 1 } { 3 }\).
2. (A) Show that the vector \(\left( \begin{array} { r } 1 \\ - 6 \\ 0 \end{array} \right)\) is normal to the plane DHC.
   (B) Hence find the cartesian equation of this plane.
   (C) Given that G lies in the plane DHC , find \(b\) and the length FG .
3. Find the angle EFB . A straight wire joins point H to a point P which is half way between E and \(\mathrm { F } . \mathrm { Q }\) is a point two-thirds of the way along this wire, so that \(\mathrm { HQ } = 2 \mathrm { QP }\).
4. Find the height of Q above the ground.

2 Fig. 7 shows a tetrahedron ABCD . The coordinates of the vertices, with respect to axes Oxyz , are \(\mathrm { A } ( - 3,0,0 ) , \mathrm { B } ( 2,0 , - 2 ) , \mathrm { C } ( 0,4,0 )\) and \(\mathrm { D } ( 0,4,5 )\). \begin{figure}[h] \end{figure}

1. Find the length of the edges AB and AC , and the size of the angle CAB . Hence calculate the area of triangle ABC .
2. (A) Verify that \(4 \mathbf { i } - 3 \mathbf { j } + 10 \mathbf { k }\) is normal to the plane ABC .
   (B) Hence find the equation of this plane.
3. Write down a vector equation for the line through D perpendicular to the plane ABC . Hence find the point of intersection of this line with the plane ABC . The volume of a tetrahedron is \(\frac { 1 } { 3 } \times\) area of base × height.
4. Find the volume of the tetrahedron ABCD .
5. Find a vector equation of the line \(l\) joining the points \(( 0,1,3 )\) and \(( - 2,2,5 )\).
6. Find the point of intersection of the line \(l\) with the plane \(x + 3 y + 2 z = 4\).
7. Find the acute angle between the line \(l\) and the normal to the plane.

2 A piece of cloth ABDC is attached to the tops of vertical poles \(\mathrm { AE } , \mathrm { BF } , \mathrm { DG }\) and CH , where \(\mathrm { E } , \mathrm { F } , \mathrm { G }\) and H are at ground level (see Fig. 7). Coordinates are as shown, with lengths in metres. The length of pole DG is \(k\) metres. \begin{figure}[h] \end{figure}

1. Write down the vectors \(\overrightarrow { \mathrm { AB } }\) and \(\overrightarrow { \mathrm { AC } }\). Hence calculate the angle BAC .
2. Verify that the equation of the plane ABC is \(x + y - 2 z + d = 0\), where \(d\) is a constant to be determined. Calculate the acute angle the plane makes with the horizontal plane.
3. Given that \(\mathrm { A } , \mathrm { B } , \mathrm { D }\) and C are coplanar, show that \(k = 3\). Hence show that ABDC is a trapezium, and find the ratio of CD to AB .

3 A straight pipeline AB passes through a mountain. With respect to axes \(\mathrm { O } x y z\), with \(\mathrm { O } x\) due East, Oy 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 \(\left. \overrightarrow { \mathrm { AB } } = \begin{array} { l } 300 \\ 100 \\ 100 \end{array} \right)\) and find the length of the pipeline.
   [0pt] [3]
2. Write down a vector equation of the line AB , and calculate the angle it makes with the vertical.
   [0pt] [6]
   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.

4 When a light ray passes from air to glass, it is deflected through an angle. The light ray ABC starts at point \(\mathrm { A } ( 1,2,2 )\), and enters a glass object at point \(\mathrm { B } ( 0,0,2 )\). 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 \(\mathbf { n }\). \begin{figure}[h] \end{figure}

1. Find the vector \(\overrightarrow { \mathrm { AB } }\) and a vector equation of the line AB . 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.
2. Write down the normal vector \(\mathbf { n }\), and hence calculate \(\theta\), giving your answer in degrees. The line BC has vector equation \(\mathbf { r } = \left( \begin{array} { l } 0 \\ 0 \\ 2 \end{array} \right) + \mu \left( \begin{array} { l } - 2 \\ - 2 \\ - 1 \end{array} \right)\). This line makes an acute angle \(\phi\) with the
   normal to the plane. normal to the plane.
3. Show that \(\phi = 45 ^ { \circ }\).
4. Snell's Law states that \(\sin \theta = k \sin \phi\), where \(k\) is a constant called the refractive index. Find \(k\). The light ray leaves the glass object through a plane with equation \(x + z = - 1\). Units are centimetres.
5. 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.

1 The points \(\mathrm { A } , \mathrm { B }\) and C have coordinates \(\mathrm { A } ( 3,2 , - 1 ) , \mathrm { B } ( - 1,1,2 )\) and \(\mathrm { C } ( 10,5 , - 5 )\), relative to the origin O . Show that \(\overrightarrow { \mathrm { OC } }\) can be written in the form \(\lambda \overrightarrow { \mathrm { OA } } + \mu \overrightarrow { \mathrm { OB } }\), where \(\lambda\) and \(\mu\) are to be determined. What can you deduce about the points \(\mathrm { O } , \mathrm { A } , \mathrm { B }\) and C from the fact that \(\overrightarrow { \mathrm { OC } }\) can be expressed as a combination of \(\overrightarrow { \mathrm { OA } }\) and \(\overrightarrow { \mathrm { OB } }\) ?

2 Vectors \(\mathbf { a }\) and \(\mathbf { b }\) are given by \(\mathbf { a } = 2 \mathbf { i } + \mathbf { j } - \mathbf { k }\) and \(\mathbf { b } = 4 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }\).
Find constants \(\lambda\) and \(\mu\) such that \(\lambda \mathbf { a } + \mu \mathbf { b } = 4 \mathbf { j } - 3 \mathbf { k }\).

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. \begin{figure}[h] \end{figure} Figure 1 shows a regular hexagon \(O P Q R S T\).
The vectors \(\mathbf { p }\) and \(\mathbf { q }\) are defined by \(\mathbf { p } = \overrightarrow { O P }\) and \(\mathbf { q } = \overrightarrow { O Q }\) Forces, in Newtons, \(\mathbf { F } _ { P } = ( \overrightarrow { O P } ) , \mathbf { F } _ { Q } = 2 \times ( \overrightarrow { O Q } ) , \mathbf { F } _ { R } = 3 \times ( \overrightarrow { O R } ) , \mathbf { F } _ { S } = 4 \times ( \overrightarrow { O S } )\) and \(\mathbf { F } _ { T } = 5 \times ( \overrightarrow { O T } )\) are applied to a particle.

1. Find, in terms of \(\mathbf { p }\) and \(\mathbf { q }\), the resultant force on the particle. The magnitude of the acceleration of the particle due to these forces is \(13 \mathrm {~ms} ^ { - 2 }\) Given that the mass of the particle is 3 kg ,
2. find \(| \mathbf { p } |\) \includegraphics[max width=\textwidth, alt={}, center]{71cd126f-1c7d-4e37-a26d-7ff98a74fd79-04_2255_56_310_1980}

5.The lines \(L _ { 1 }\) and \(L _ { 2 }\) have vector equations \(L _ { 1 } : \quad \mathbf { r } = - 2 \mathbf { i } + 11.5 \mathbf { j } + \lambda ( 3 \mathbf { i } - 4 \mathbf { j } - \mathbf { k } )\), \(L _ { 2 } : \quad \mathbf { r } = 11.5 \mathbf { i } + 3 \mathbf { j } + 8.5 \mathbf { k } + \mu ( 7 \mathbf { i } + 8 \mathbf { j } - 11 \mathbf { k } )\),
where \(\lambda\) and \(\mu\) are parameters.

1. Show that \(L _ { 1 }\) and \(L _ { 2 }\) do not intersect.
2. Show that the vector \(( 2 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } )\) is perpendicular to both \(L _ { 1 }\) and \(L _ { 2 }\) . The point \(A\) lies on \(L _ { 1 }\) ,the point \(B\) lies on \(L _ { 2 }\) and \(A B\) is perpendicular to both \(L _ { 1 }\) and \(L _ { 2 }\) .
3. Find the position vector of the point \(A\) and the position vector of the point \(B\) .
   (8) \includegraphics[max width=\textwidth, alt={}, center]{0df09d8a-7478-4679-b117-128ee226db6a-4_554_1017_404_571} Figure 1 shows a sketch of part of the curve \(C\) with equation $$y = \sin ( \ln x ) , \quad x \geq 1 .$$ The point \(Q\) ,on \(C\) ,is a maximum.

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.Relative to a fixed origin \(O\) the points \(A , B\) and \(C\) have position vectors $$\mathbf { a } = - \mathbf { i } + \frac { 4 } { 3 } \mathbf { j } + 7 \mathbf { k } , \quad \mathbf { b } = 4 \mathbf { i } + \frac { 4 } { 3 } \mathbf { j } + 2 \mathbf { k } \text { and } \mathbf { c } = 6 \mathbf { i } + \frac { 16 } { 3 } \mathbf { j } + 2 \mathbf { k } \text { respectively. }$$

1. Find the cosine of angle \(A B C\) . The quadrilateral \(A B C D\) is a kite \(K\) .
2. Find the area of \(K\) . A circle is drawn inside \(K\) so that it touches each of the 4 sides of \(K\) .
3. Find the radius of the circle,giving your answer in the form \(p \sqrt { } ( q ) - q \sqrt { } ( p )\) ,where \(p\) and \(q\) are positive integers.
4. Find the position vector of the point \(D\) .
   (Total 18 marks)

3.The points \(A , B , C , D\) and \(E\) are five of the vertices of a rectangular cuboid and \(A E\) is a diagonal of the cuboid.With respect to a fixed origin \(O\) ,the position vectors of \(A , B , C\) and \(D\) are \(\mathbf { a , b , c }\) and \(\mathbf{d}\) respectively,where $$\mathbf { a } = \left( \begin{array} { c } 1 \\ 2 \\ - 1 \end{array} \right) , \quad \mathbf { b } = \left( \begin{array} { c } 0 \\ - 3 \\ - 8 \end{array} \right) , \quad \mathbf { c } = \left( \begin{array} { c } 4 \\ - 1 \\ - 10 \end{array} \right)$$

\text { and } \mathbf { d } = \left( \begin{array} { c } - 4
2
- 11 \end{array} \right)$$

1. Find the position vector of \(E\) . The volume of a tetrahedron is given by the formula $$\text { volume } = \frac { 1 } { 3 } ( \text { area of base } ) \times ( \text { height } )$$
2. Find the volume of the tetrahedron \(A B C D\) . 4.(a)Given that \(x > 0 , y > 0 , x \neq 1\) and \(n > 0\) ,show that $$\log _ { x } y = \log _ { x ^ { n } } y ^ { n }$$

(b)Solve the following,leaving your answers in the form \(2 ^ { p }\) ,where \(p\) is a rational number.

1. \(\log _ { 2 } u + \log _ { 4 } u ^ { 2 } + \log _ { 8 } u ^ { 3 } + \log _ { 16 } u ^ { 4 } = 5\)
2. \(\log _ { 2 } v + \log _ { 4 } v + \log _ { 8 } v + \log _ { 16 } v = 5\)
3. \(\log _ { 4 } w ^ { 2 } + \frac { 3 \log _ { 8 } 64 } { \log _ { 2 } w } = 5\)

3. \begin{figure}[h] \end{figure} Figure 1 shows a regular pentagon \(O A B C D\). The vectors \(\mathbf { p }\) and \(\mathbf { q }\) are defined by \(\mathbf { p } = \overrightarrow { O A }\) and \(\mathbf { q } = \overrightarrow { O D }\) respectively. Let \(k\) be the number such that \(\overrightarrow { D B } = k \overrightarrow { O A }\).

1. Write down \(\overrightarrow { A C }\) in terms of \(\mathbf { p } , \mathbf { q }\) and \(k\) as appropriate.
2. Show that \(\overrightarrow { C D } = - \mathbf { p } - \frac { 1 } { k } \mathbf { q }\)
3. Hence find the value of \(k\) By considering triangle \(D B C\), or otherwise,
4. find the exact value of \(\sin 54 ^ { \circ }\)

4 The line \(l\) has equations \(\frac { x - 1 } { 2 } = \frac { y - 1 } { 3 } = \frac { z + 1 } { 2 }\) and the point \(A\) is ( \(7,3,7\) ). \(M\) is the point where the perpendicular from \(A\) meets \(l\).

1. Find, in either order, the coordinates of \(M\) and the perpendicular distance from \(A\) to \(l\).
2. Find the coordinates of the point \(B\) on \(A M\) such that \(\overrightarrow { A B } = 3 \overrightarrow { B M }\).

5 \includegraphics[max width=\textwidth, alt={}, center]{b5d85e48-0d5a-4edf-bf58-eba4f8d28d3d-2_425_680_1302_689} In the diagram the points \(A\) and \(B\) have position vectors \(\mathbf { a }\) and \(\mathbf { b }\) with respect to the origin \(O\). Given that \(| \mathbf { a } | = 3 , | \mathbf { b } | = 4\) and \(\mathbf { a . b } = 6\), find

1. the angle \(A O B\),
2. \(| \mathbf { a } - \mathbf { b } |\).

10 Lines \(l _ { 1 }\) and \(l _ { 2 }\) have vector equations $$\mathbf { r } = - \mathbf { i } + 2 \mathbf { j } + 7 \mathbf { k } + t ( 2 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 2 \mathbf { i } + 9 \mathbf { j } - 4 \mathbf { k } + s ( \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k } )$$ respectively. The point \(A\) has coordinates ( \(- 3,0,6\) ) relative to the origin \(O\).

1. Show that \(A\) lies on \(l _ { 1 }\) and that \(O A\) is perpendicular to \(l _ { 1 }\).
2. Show that the line through \(O\) and \(A\) intersects \(l _ { 2 }\).
3. Given that the point of intersection in part (ii) is \(B\), find the ratio \(| \overrightarrow { O A } | : | \overrightarrow { B A } |\). \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}

7 Points \(A ( 2,2,5 ) , B ( 1 , - 1 , - 4 ) , C ( 3,3,10 )\) and \(D ( 8,6,3 )\) are the vertices of a pyramid with a triangular base.

1. Calculate the lengths \(A B\) and \(A C\), and the angle \(B A C\).
2. Show that \(\overrightarrow { A D }\) is perpendicular to both \(\overrightarrow { A B }\) and \(\overrightarrow { A C }\).
3. Calculate the volume of the pyramid \(A B C D\).
   [0pt] [The volume of the pyramid is \(V = \frac { 1 } { 3 } \times\) base area × perpendicular height.]

2 The points \(O ( 0,0,0 ) , A ( 2,8,2 ) , B ( 5,5,8 )\) and \(C ( 3 , - 3,6 )\) form a parallelogram \(O A B C\). Use a scalar product to find the acute angle between the diagonals of this parallelogram.

8 The points \(A\) and \(B\) have position vectors relative to the origin \(O\) given by $$\overrightarrow { O A } = \left( \begin{array} { c } 3 \sin \alpha \\ 2 \cos \alpha \\ - 1 \end{array} \right) \text { and } \overrightarrow { O B } = \left( \begin{array} { c } 2 \cos \alpha \\ 4 \sin \alpha \\ 3 \end{array} \right)$$ where \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). It is given that \(\overrightarrow { O A }\) and \(\overrightarrow { O B }\) are perpendicular.

1. Calculate the two possible values of \(\alpha\).
2. Calculate the area of triangle \(O A B\) for the smaller value of \(\alpha\) from part (i).

3 Vectors \(\mathbf { a }\) and \(\mathbf { b }\) are given by \(\mathbf { a } = 2 \mathbf { i } + \mathbf { j } - \mathbf { k }\) and \(\mathbf { b } = 4 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }\).
Find constants \(\lambda\) and \(\mu\) such that \(\lambda \mathbf { a } + \mu \mathbf { b } = 4 \mathbf { j } - 3 \mathbf { k }\).

4 The points A , B and C have coordinates \(( 1,3 , - 2 ) , ( - 1,2 , - 3 )\) and \(( 0 , - 8,1 )\) respectively.

1. Find the vectors \(\overrightarrow { \mathrm { AB } }\) and \(\overrightarrow { \mathrm { AC } }\).
2. Show that the vector \(2 \mathbf { i } - \mathbf { j } - 3 \mathbf { k }\) is perpendicular to the plane ABC . Hence find the equation of the plane ABC .