Volume of tetrahedron using scalar triple product

10 With respect to the origin \(O\), the points \(A , B , C\) and \(D\) have position vectors given by $$\overrightarrow { O A } = \left( \begin{array} { r } 3
- 1
2 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 1
2
- 3 \end{array} \right) , \quad \overrightarrow { O C } = \left( \begin{array} { r } 1
- 2
5 \end{array} \right) \quad \text { and } \quad \overrightarrow { O D } = \left( \begin{array} { r } 5
- 6

6. The coordinates of the points \(A , B\) and \(C\) relative to a fixed origin \(O\) are ( \(1,2,3\) ), \(( - 1,3,4 )\) and \(( 2,1,6 )\) respectively. The plane \(\Pi\) contains the points \(A , B\) and \(C\).

1. Find a cartesian equation of the plane \(\Pi\). The point \(D\) has coordinates \(( k , 4,14 )\) where \(k\) is a positive constant.
   Given that the volume of the tetrahedron \(A B C D\) is 6 cubic units,
2. find the value of \(k\).
   

   VIIIV SIHI NI IIIIM I I O N OA

   VI4V SIHI NI JIIIM ION OC

   VJYV SIHI NI JIIIM ION OO

9 With respect to a fixed origin \(O\), the points \(A ( - 1,5,1 ) , B ( 1,0,3 ) , C ( 2 , - 1,2 )\) and \(D ( 3,6 , - 1 )\) are the vertices of a tetrahedron.

1. Find the volume of the tetrahedron \(A B C D\). The plane \(\Pi\) contains the points \(A , B\) and \(C\).
2. Find a cartesian equation of \(\Pi\). The point \(T\) lies on the plane \(\Pi\). The line \(D T\) is perpendicular to \(\Pi\).
3. Find the exact coordinates of the point \(T\).

1. The coordinates of the points \(A , B\) and \(C\) relative to a fixed origin \(O\) are \(( 1,2,3 )\),

The point \(D\) has coordinates \(( k , 4,14 )\) where \(k\) is a positive constant.
Given that the volume of the tetrahedron \(A B C D\) is 6 cubic units,
(b) find the value of \(k\). \section*{\(( - 1,3,4 )\) and \(( 2,1,6 )\) respectively. The plane \(\Pi\) contains the points \(A , B\) and \(C\).
(a) Find a cartesian equation of the plane \(\Pi\).
6. \(( - 1,3,4 )\) and \(( 2,1,6 )\) respectively. The plane (a) Find a cartesian equation of the plane \(\Pi\).}

2. \begin{figure}[h] \end{figure} The points \(A , B\) and \(C\) have position vectors \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\) respectively, relative to a fixed origin \(O\), as shown in Figure 1. It is given that $$\mathbf { a } = \mathbf { i } + \mathbf { j } , \quad \mathbf { b } = 3 \mathbf { i } - \mathbf { j } + \mathbf { k } \quad \text { and } \quad \mathbf { c } = 2 \mathbf { i } + \mathbf { j } - \mathbf { k } .$$ Calculate

1. \(\mathbf { b } \times \mathbf { c }\),
2. a.(b \(\times \mathbf { c ) }\),
3. the area of triangle \(O B C\),
4. the volume of the tetrahedron \(O A B C\).

2. \begin{figure}[h] \end{figure} The points \(A , B\) and \(C\) have position vectors \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\) respectively, relative to a fixed origin \(O\), as shown in Figure 1. It is given that $$\mathbf { a } = \mathbf { i } + \mathbf { j } , \quad \mathbf { b } = 3 \mathbf { i } - \mathbf { j } + \mathbf { k } \quad \text { and } \quad \mathbf { c } = 2 \mathbf { i } + \mathbf { j } - \mathbf { k } .$$ Calculate

1. \(\mathbf { b } \times \mathbf { c }\),
2. a.(b \(\times \mathbf { c ) }\),
3. the area of triangle \(O B C\),
4. the volume of the tetrahedron \(O A B C\).

8. The points \(A , B , C\), and \(D\) have position vectors $$\mathbf { a } = 2 \mathbf { i } + \mathbf { k } , \mathrm { b } = \mathbf { i } + 3 \mathbf { j } , \mathbf { c } = \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k } , \mathbf { d } = 4 \mathbf { j } + \mathbf { k }$$ respectively.

1. Find \(\overrightarrow { A B } \times \overrightarrow { A C }\) and hence find the area of triangle \(A B C\).
2. Find the volume of the tetrahedron \(A B C D\).
3. Find the perpendicular distance of \(D\) from the plane containing \(A , B\) and \(C\).

1 A tetrahedron ABCD has vertices \(\mathrm { A } ( - 3,5,2 ) , \mathrm { B } ( 3,13,7 ) , \mathrm { C } ( 7,0,3 )\) and \(\mathrm { D } ( 5,4,8 )\).

1. Find the vector product \(\overrightarrow { \mathrm { AB } } \times \overrightarrow { \mathrm { AC } }\), and hence find the equation of the plane ABC .
2. Find the shortest distance from \(D\) to the plane \(A B C\).
3. Find the shortest distance between the lines AB and CD .
4. Find the volume of the tetrahedron ABCD . The plane \(P\) with equation \(3 x - 2 z + 5 = 0\) contains the point B , and meets the lines AC and AD at E and F respectively.
5. Find \(\lambda\) and \(\mu\) such that \(\overrightarrow { \mathrm { AE } } = \lambda \overrightarrow { \mathrm { AC } }\) and \(\overrightarrow { \mathrm { AF } } = \mu \overrightarrow { \mathrm { AD } }\). Deduce that E is between A and C , and that F is between A and D.
6. Hence, or otherwise, show that \(P\) divides the tetrahedron ABCD into two parts having volumes in the ratio 4 to 17.

1 Three points have coordinates \(\mathrm { A } ( - 3,12 , - 7 ) , \mathrm { B } ( - 2,6,9 ) , \mathrm { C } ( 6,0 , - 10 )\). The plane \(P\) passes through the points \(\mathrm { A } , \mathrm { B }\) and C .

1. Find the vector product \(\overrightarrow { \mathrm { AB } } \times \overrightarrow { \mathrm { AC } }\). Hence or otherwise find an equation for the plane \(P\) in the form \(a x + b y + c z = d\). The plane \(Q\) has equation \(6 x + 3 y + 2 z = 32\). The perpendicular from A to the plane \(Q\) meets \(Q\) at the point D. The planes \(P\) and \(Q\) intersect in the line \(L\).
2. Find the distance AD .
3. Find an equation for the line \(L\).
4. Find the shortest distance from A to the line \(L\).
5. Find the volume of the tetrahedron ABCD .

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

6
1
1 \end{array} \right)$$

1. Find the vectors \(\overrightarrow { A B } , \overrightarrow { A C } , \overrightarrow { A D } , \overrightarrow { B C } , \overrightarrow { B D }\) and \(\overrightarrow { C D }\).
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 \(E\). 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.\\ 6. (i) A curve with equation \(y = \mathrm { f } ( x )\) has \(\mathrm { f } ( x ) \geqslant 0\) for \(x \geqslant a\) and $$A = \int _ { a } ^ { b } \mathrm { f } ( x ) \mathrm { d } x \quad \text { and } \quad V = \pi \int _ { a } ^ { b } [ \mathrm { f } ( x ) ] ^ { 2 } \mathrm {~d} x$$ where \(a\) and \(b\) are constants with \(b > a\). Use integration by substitution to show that for the positive constants \(r\) and \(h\) $$\pi \int _ { a + h } ^ { b + h } [ r + \mathrm { f } ( x - h ) ] ^ { 2 } \mathrm {~d} x = \pi r ^ { 2 } ( b - a ) + 2 \pi r A + V$$ (ii) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{65b3fb2e-c603-48a0-b4ad-0f78603c203c-5_492_1038_799_504} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} Figure 1 shows part of the curve \(C\) with equation \(y = 4 + \frac { 2 } { \sqrt { 3 } \cos x + \sin x }\) This curve has asymptotes \(x = m\) and \(x = n\) and crosses the \(y\)-axis at \(( 0 , p )\).

8
- 6 \end{array} \right) , \quad \mathbf { c } = \left( \begin{array} { l } 2
5
3 \end{array} \right) , \quad \mathbf { d } = \left( \begin{array} { l } 6
1
1 \end{array} \right)$$

1. Find the vectors \(\overrightarrow { A B } , \overrightarrow { A C } , \overrightarrow { A D } , \overrightarrow { B C } , \overrightarrow { B D }\) and \(\overrightarrow { C D }\).
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 \(E\). 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.\\ 6. (i) A curve with equation \(y = \mathrm { f } ( x )\) has \(\mathrm { f } ( x ) \geqslant 0\) for \(x \geqslant a\) and $$A = \int _ { a } ^ { b } \mathrm { f } ( x ) \mathrm { d } x \quad \text { and } \quad V = \pi \int _ { a } ^ { b } [ \mathrm { f } ( x ) ] ^ { 2 } \mathrm {~d} x$$ where \(a\) and \(b\) are constants with \(b > a\). Use integration by substitution to show that for the positive constants \(r\) and \(h\) $$\pi \int _ { a + h } ^ { b + h } [ r + \mathrm { f } ( x - h ) ] ^ { 2 } \mathrm {~d} x = \pi r ^ { 2 } ( b - a ) + 2 \pi r A + V$$ (ii) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{65b3fb2e-c603-48a0-b4ad-0f78603c203c-5_492_1038_799_504} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} Figure 1 shows part of the curve \(C\) with equation \(y = 4 + \frac { 2 } { \sqrt { 3 } \cos x + \sin x }\)\\ This curve has asymptotes \(x = m\) and \(x = n\) and crosses the \(y\)-axis at \(( 0 , p )\).
4. Find the value of \(p\), the value of \(m\) and the value of \(n\).
5. Show that the equation of \(C\) can be written in the form \(y = r + \mathrm { f } ( x - h )\) and specify the function f and the constants \(r\) and \(h\). The region bounded by \(C\), the \(x\)-axis and the lines \(x = \frac { \pi } { 6 }\) and \(x = \frac { \pi } { 3 }\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
6. Find the volume of the solid formed.\\ 7. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{65b3fb2e-c603-48a0-b4ad-0f78603c203c-6_631_974_201_548} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} A circular tower stands in a large horizontal field of grass．A goat is attached to one end of a string and the other end of the string is attached to the fixed point \(O\) at the base of the tower．Taking the point \(O\) as the origin（ 0,0 ），the centre of the base of the tower is at the point \(T ( 0,1 )\) ．The radius of the base of the tower is 1 ．The string has length \(\pi\) and you may ignore the size of the goat．The curve \(C\) represents the edge of the region that the goat can reach as shown in Figure 2.\\ （a）Write down the equation of \(C\) for \(y < 0\) ． When the goat is at the point \(G ( x , y )\) ，with \(x > 0\) and \(y > 0\) ，as shown in Figure 2 ，the string lies along \(O A G\) where \(O A\) is an arc of the circle with angle \(O T A = \theta\) radians and \(A G\) is a tangent to the circle at \(A\) ．\\ （b）With the aid of a suitable diagram show that $$\begin{aligned} & x = \sin \theta + ( \pi - \theta ) \cos \theta
   & y = 1 - \cos \theta + ( \pi - \theta ) \sin \theta \end{aligned}$$ （c）By considering \(\int y \frac { \mathrm {~d} x } { \mathrm {~d} \theta } \mathrm {~d} \theta\) ，show that the area between \(C\) ，the positive \(x\)－axis and the positive \(y\)－axis can be expressed in the form $$\int _ { 0 } ^ { \pi } u \sin u \mathrm {~d} u + \int _ { 0 } ^ { \pi } u ^ { 2 } \sin ^ { 2 } u \mathrm {~d} u + \int _ { 0 } ^ { \pi } u \sin u \cos u \mathrm {~d} u$$ （d）Show that \(\int _ { 0 } ^ { \pi } u ^ { 2 } \sin ^ { 2 } u \mathrm {~d} u = \frac { \pi ^ { 3 } } { 6 } + \int _ { 0 } ^ { \pi } u \sin u \cos u \mathrm {~d} u\) （e）Hence find the area of grass that can be reached by the goat．

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 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)$$ （a）Find the position vector of \(E\) ． The volume of a tetrahedron is given by the formula $$\text { volume } = \frac { 1 } { 3 } ( \text { area of base } ) \times ( \text { height } )$$ （b）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．
（i） \(\log _ { 2 } u + \log _ { 4 } u ^ { 2 } + \log _ { 8 } u ^ { 3 } + \log _ { 16 } u ^ { 4 } = 5\) （ii） \(\log _ { 2 } v + \log _ { 4 } v + \log _ { 8 } v + \log _ { 16 } v = 5\) （iii） \(\log _ { 4 } w ^ { 2 } + \frac { 3 \log _ { 8 } 64 } { \log _ { 2 } w } = 5\)

The points \(A , B , C\) and \(D\) have coordinates as follows: $$A ( 2,1 , - 2 ) , \quad B ( 4,1 , - 1 ) , \quad C ( 3 , - 2 , - 1 ) \quad \text { and } \quad D ( 3,6,2 ) .$$ The plane \(\Pi _ { 1 }\) passes through the points \(A , B\) and \(C\). Find a cartesian equation of \(\Pi _ { 1 }\). Find the area of triangle \(A B C\) and hence, or otherwise, find the volume of the tetrahedron \(A B C D\).
[0pt] [The volume of a tetrahedron is \(\frac { 1 } { 3 } \times\) area of base × perpendicular height.]
The plane \(\Pi _ { 2 }\) passes through the points \(A , B\) and \(D\). Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\). \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. }

2 The position vectors of points \(A , B , C\), relative to the origin \(O\), are \(\mathbf { a } , \mathbf { b } , \mathbf { c }\), where $$\mathbf { a } = 3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k } , \quad \mathbf { b } = 4 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k } , \quad \mathbf { c } = 3 \mathbf { i } - \mathbf { j } - \mathbf { k }$$ Find \(\mathbf { a } \times \mathbf { b }\) and deduce the area of the triangle \(O A B\). Hence find the volume of the tetrahedron \(O A B C\), given that the volume of a tetrahedron is \(\frac { 1 } { 3 } \times\) area of base × perpendicular height.

16 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 \(A B C D E F\) with parallel triangular ends \(A B C\) and \(D E F\), and a rectangular base \(A C F D\). He uses the metre as the unit of length. \includegraphics[max width=\textwidth, alt={}, center]{21084ed7-43f8-47c6-80c2-930ccf340d37-22_510_766_484_776} The coordinates of \(B , C\) and \(D\) are ( \(3,1,11\) ), ( \(9,3,4\) ) and ( \(- 4,12,4\) ) respectively.
He uses the equation \(x - 3 y = 0\) for the plane \(A B C\).
He uses \(\left[ \mathbf { r } - \left( \begin{array} { c } - 4 \\ 12 \\ 4 \end{array} \right) \right] \times \left( \begin{array} { c } 4 \\ - 12 \\ 0 \end{array} \right) = \left( \begin{array} { l } 0 \\ 0 \\ 0 \end{array} \right)\) for the equation of the line \(A D\).
Find the volume of the space enclosed inside this section of the roof.
[0pt] [9 marks]

8 The points \(P , Q\) and \(R\) have coordinates \(( 0,2,3 ) , ( 2,0,1 )\) and \(( 1,3,0 )\) respectively.
The acute angle between the line segments \(P Q\) and \(P R\) is \(\theta\).

1. Show that \(\sin \theta = \frac { 2 } { 11 } \sqrt { 22 }\). The triangle \(P Q R\) lies in the plane \(\Pi\).
2. Determine an equation for \(\Pi\), giving your answer in the form \(\mathrm { ax } + \mathrm { by } + \mathrm { cz } = \mathrm { d }\), where \(a , b , c\) and \(d\) are integers. The point \(S\) has coordinates \(( 5,3 , - 1 )\).
3. By finding the shortest distance between \(S\) and the plane \(\Pi\), show that the volume of the tetrahedron \(P Q R S\) is \(\frac { 14 } { 3 }\).
   [0pt] [The volume of a tetrahedron is \(\frac { 1 } { 3 } \times\) area of base × perpendicular height] The tetrahedron \(P Q R S\) is transformed to the tetrahedron \(\mathrm { P } ^ { \prime } \mathrm { Q } ^ { \prime } \mathrm { R } ^ { \prime } \mathrm { S } ^ { \prime }\) by a rotation about the \(y\)-axis.
   The \(x\)-coordinate of \(S ^ { \prime }\) is \(2 \sqrt { 2 }\).
4. By using the matrix for a rotation by angle \(\theta\) about the \(y\)-axis, as given in the Formulae Booklet, determine in exact form the possible coordinates of \(R ^ { \prime }\).

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of a solid doorstop made of wood. The doorstop is modelled as a tetrahedron. Relative to a fixed origin \(O\), the vertices of the tetrahedron are \(A ( 2,1,4 )\), \(B ( 6,1,2 ) , C ( 4,10,3 )\) and \(D ( 5,8 , d )\), where \(d\) is a positive constant and the units are in centimetres.

1. Find the area of the triangle \(A B C\). Given that the volume of the doorstop is \(21 \mathrm {~cm} ^ { 3 }\)
2. find the value of the constant \(d\).

5. \begin{figure}[h] \end{figure} Figure 3 shows a solid display stand with parallel triangular faces \(A B C\) and \(D E F\). Triangle \(D E F\) is similar to triangle \(A B C\). With respect to a fixed origin \(O\), the points \(A , B\) and \(C\) have coordinates ( \(3 , - 3,1\) ), ( \(- 5,3,3\) ) and ( \(1,7,5\) ) respectively and the points \(D , E\) and \(F\) have coordinates ( \(2 , - 1,8\) ), ( \(- 2,2,9\) ) and ( \(1,4,10\) ) respectively. The units are in centimetres.

1. Show that the area of the triangular face \(D E F\) is \(\frac { 1 } { 2 } \sqrt { 339 } \mathrm {~cm} ^ { 2 }\)
2. Find, in \(\mathrm { cm } ^ { 3 }\), the exact volume of the display stand.

1. With respect to a fixed origin \(O\), the points \(A\), \(B\) and \(C\) have position vectors given by

$$\overrightarrow { O A } = 18 \mathbf { i } - 14 \mathbf { j } - 2 \mathbf { k } \quad \overrightarrow { O B } = - 7 \mathbf { i } - 5 \mathbf { j } + 3 \mathbf { k } \quad \overrightarrow { O C } = - 2 \mathbf { i } - 9 \mathbf { j } - 6 \mathbf { k }$$ The points \(O , A , B\) and \(C\) form the vertices of a tetrahedron.

1. Show that the area of the triangular face \(A B C\) of the tetrahedron is 108 to 3 significant figures.
2. Find the volume of the tetrahedron. An oak wood block is made in the shape of the tetrahedron, with centimetres taken for the units. The density of oak is \(0.85 \mathrm {~g} \mathrm {~cm} ^ { - 3 }\)
3. Determine the mass of the block, giving your answer in kg.

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of a solid sculpture made of glass and concrete. The sculpture is modelled as a parallelepiped. The sculpture is made up of a concrete solid in the shape of a tetrahedron, shown shaded in Figure 1, whose vertices are \(\mathrm { O } ( 0,0,0 ) , \mathrm { A } ( 2,0,0 ) , \mathrm { B } ( 0,3,1 )\) and \(\mathrm { C } ( 1,1,2 )\), where the units are in metres. The rest of the solid parallelepiped is made of glass which is glued to the concrete tetrahedron.

1. Find the surface area of the glued face of the tetrahedron.
2. Find the volume of glass contained in this parallelepiped.
3. Give a reason why the volume of concrete predicted by this model may not be an accurate value for the volume of concrete that was used to make the sculpture. \section*{Q uestion 4 continued}

1. With respect to a fixed origin \(O\), the points \(A\), \(B\) and \(C\) have coordinates \(( 3,4,5 ) , ( 10 , - 1,5 )\) and ( \(4,7 , - 9\) ) respectively.

The plane \(\Pi\) has equation \(4 x - 8 y + z = 2\) The line segment \(A B\) meets the plane \(\Pi\) at the point \(P\) and the line segment \(B C\) meets the plane \(\Pi\) at the point \(Q\).

1. Show that, to 3 significant figures, the area of quadrilateral \(A P Q C\) is 38.5 The point \(D\) has coordinates \(( k , 4 , - 1 )\), where \(k\) is a constant.
   Given that the vectors \(\overrightarrow { A B } , \overrightarrow { A C }\) and \(\overrightarrow { A D }\) form three edges of a parallelepiped of volume 226
2. find the possible values of the constant \(k\).