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 $$\overrightarrow { O A } = \left( \begin{array} { r } 3 \\ - 1 \\ 2 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 1 \\ 2 \\ - 3 \end{array} \right) , \quad \overrightarrow { O C } = \left( \begin{array} { r } 1 \\ - 2 \\ 5 \end{array} \right) \quad \text { and } \quad \overrightarrow { O D } = \left( \begin{array} { r } 5 \\ - 6 \\ 11 \end{array} \right)$$

1. Find the obtuse angle between the vectors \(\overrightarrow { O A }\) and \(\overrightarrow { O B }\). \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) The line \(l\) passes through the points \(A\) and \(B\).
2. Find a vector equation for the line \(l\). \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
3. Find the position vector of the point of intersection of the line \(l\) and the line passing through \(C\) and \(D\). \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

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 \(k\).
   

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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 \(T\).

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 \(O A B C\).

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 .

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 $$\log _ { x } y = \log _ { x ^ { n } } y ^ { n }$$
3. 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. \(\log _ { 4 } w ^ { 2 } + \frac { 3 \log _ { 8 } 64 } { \log _ { 2 } w } = 5\)

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 \(\Pi _ { 2 }\).

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.

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 \(k\).

1. A tetrahedron has vertices \(A ( 1,2,1 ) , B ( 0,1,0 ) , C ( 2,1,3 )\) and \(D ( 10,5,5 )\).

Find

1. a Cartesian equation of the plane \(A B C\).
2. the volume of the tetrahedron \(A B C D\). The plane \(\Pi\) has equation \(2 x - 3 y + 3 = 0\) The point \(E\) lies on the line \(A C\) and the point \(F\) lies on the line \(A D\).
   Given that \(\Pi\) contains the point \(B\), the point \(E\) and the point \(F\),
3. find the value of \(k\) such that \(\overrightarrow { A E } = k \overrightarrow { A C }\). Given that \(\overrightarrow { A F } = \frac { 1 } { 9 } \overrightarrow { A D }\)
4. show that the volume of the tetrahedron \(A B C D\) is 45 times the volume of the tetrahedron \(A B E F\).

4. $$\mathbf { a } = \left( \begin{array} { r } - 3 \\ 1 \\ 4 \end{array} \right) , \quad \mathbf { b } = \left( \begin{array} { r } 5 \\ - 2 \\ 9 \end{array} \right) , \quad \mathbf { c } = \left( \begin{array} { r } 8 \\ - 4 \\ 3 \end{array} \right)$$ The points \(A , B\) and \(C\) with position vectors \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\) ,respectively,are 3 vertices of a cube.

1. Find the volume of the cube. The points \(P , Q\) and \(R\) are vertices of a second cube with \(\overrightarrow { P Q } = \left( \begin{array} { l } 3 \\ 4 \\ \alpha \end{array} \right) , \overrightarrow { P R } = \left( \begin{array} { l } 7 \\ 1 \\ 0 \end{array} \right)\) and \(\alpha\) a positive constant.
2. Given that angle \(Q P R = 60 ^ { \circ }\) ,find the value of \(\alpha\) .
3. Find the length of a diagonal of the second cube.

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 $$\mathbf { a } = \mathbf { i } + \mathbf { j } , \quad \mathbf { b } = \mathbf { 3 i } - \mathbf { j } + \mathbf { k } \quad \text { and } \quad \mathbf { c } = \mathbf { 2 i } + \mathbf { j } - \mathbf { k } .$$ Calculate

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

1. With respect to a fixed origin \(O\) the point \(A\) has coordinates \(( 3,6,5 )\) and the line \(l\) has equation

$$( \mathbf { r } - ( 12 \mathbf { i } + 30 \mathbf { j } + 39 \mathbf { k } ) ) \times ( 7 \mathbf { i } + 13 \mathbf { j } + 24 \mathbf { k } ) = \mathbf { 0 }$$ The points \(B\) and \(C\) lie on \(l\) such that \(A B = A C = 15\) Given that \(A\) does not lie on \(l\) and that the \(x\) coordinate of \(B\) is negative,

1. determine the coordinates of \(B\) and the coordinates of \(C\)
2. Hence determine a Cartesian equation of the plane containing the points \(A , B\) and \(C\) The point \(D\) has coordinates \(( - 2,1 , \alpha )\), where \(\alpha\) is a constant.
   Given that the volume of the tetrahedron \(A B C D\) is 147
3. determine the possible values of \(\alpha\) Given that \(\alpha > 0\)
4. determine the shortest distance between the line \(l\) and the line passing through the points \(A\) and \(D\), giving your answer to 2 significant figures. \includegraphics[max width=\textwidth, alt={}, center]{c0ac1e1e-16bf-4a06-9eaa-8dcf01177722-24_2267_50_312_1980}

The points \(A\), \(B\) and \(C\) lie on the plane \(\Pi\) and, relative to a fixed origin \(O\), they have position vectors $$\mathbf{a} = 3\mathbf{i} - \mathbf{j} + 4\mathbf{k}, \quad \mathbf{b} = -\mathbf{i} + 2\mathbf{j}, \quad \mathbf{c} = 5\mathbf{i} - 3\mathbf{j} + 7\mathbf{k}$$ respectively.

1. Find \(\overrightarrow{AB} \times \overrightarrow{AC}\). [4]
2. Find an equation of \(\Pi\) in the form \(\mathbf{r} \cdot \mathbf{n} = p\). [2]

The point \(D\) has position vector \(5\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}\).

1. Calculate the volume of the tetrahedron \(ABCD\). [4]

The points \(A\), \(B\), \(C\) and \(P\) have coordinates \((a, 0, 0)\), \((0, b, 0)\), \((0, 0, c)\) and \((a, b, c)\) respectively, where \(a\), \(b\) and \(c\) are positive constants. The plane \(\Pi\) contains \(A\), \(B\) and \(C\).

1. 1. Use the scalar triple product to determine
   2. Hence show that the distance from \(P\) to \(\Pi\) is twice the distance from \(O\) to \(\Pi\). [2]
2. 1. Determine a vector which is normal to \(\Pi\). [2]
   2. Hence determine, in terms of \(a\), \(b\) and \(c\) only, the distance from \(P\) to \(\Pi\). [3]