4.04g Vector product: a x b perpendicular vector

1. Vectors \(\mathbf { u }\) and \(\mathbf { v }\) are given by

$$\mathbf { u } = 5 \mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k } \quad \text { and } \quad \mathbf { v } = a \mathbf { i } - 6 \mathbf { j } + 2 \mathbf { k }$$ where \(a\) is a constant.

1. Determine, in terms of \(a\), the vector product \(\mathbf { u } \times \mathbf { v }\) Given that
   • \(\overrightarrow { A B } = 2 \mathbf { u }\)
   • \(\overrightarrow { A C } = \mathbf { v }\)
   • the area of triangle \(A B C\) is 15
   • determine the possible values of \(a\).

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 \(k\).

4. \begin{figure}[h] \end{figure} A small aircraft is landing in a field.
In a model for the landing the aircraft travels in different straight lines before and after it lands, as shown in Figure 2. The vector \(\mathbf { v } _ { \mathbf { A } }\) is in the direction of travel of the aircraft as it approaches the field.
The vector \(\mathbf { V } _ { \mathbf { L } }\) is in the direction of travel of the aircraft after it lands.
With respect to a fixed origin, the field is modelled as the plane with equation $$x - 2 y + 25 z = 0$$ and $$\mathbf { v } _ { \mathbf { A } } = \left( \begin{array} { r } 3 \\ - 2 \\ - 1 \end{array} \right)$$

1. Write down a vector \(\mathbf { n }\) that is a normal vector to the field.
2. Show that \(\mathbf { n } \times \mathbf { v } _ { \mathbf { A } } = \lambda \left( \begin{array} { r } 13 \\ 19 \\ 1 \end{array} \right)\), where \(\lambda\) is a constant to be determined. When the aircraft lands it remains in contact with the field and travels in the direction \(\mathbf { v } _ { \mathbf { L } }\) The vector \(\mathbf { v } _ { \mathbf { L } }\) is in the same plane as both \(\mathbf { v } _ { \mathbf { A } }\) and \(\mathbf { n }\) as shown in Figure 2.
3. Determine a vector which has the same direction as \(\mathbf { V } _ { \mathbf { L } }\)
4. State a limitation of the model.

1. With respect to a fixed origin \(O\), the points \(A\) and \(B\) have coordinates \(( 2,2 , - 1 )\) and ( \(4,2 p , 1\) ) respectively, where \(p\) is a constant.

For each of the following, determine the possible values of \(p\) for which,

1. \(O B\) makes an angle of \(45 ^ { \circ }\) with the positive \(x\)-axis
2. \(\overrightarrow { O A } \times \overrightarrow { O B }\) is parallel to \(\left( \begin{array} { r } 4 \\ - p \\ 2 \end{array} \right)\)
3. the area of triangle \(O A B\) is \(3 \sqrt { 2 }\)

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

1. For non-zero vectors \(\mathbf { x }\) and \(\mathbf { y }\), explain the geometrical significance of the statement \(\mathbf { x } \times \mathbf { y } = \mathbf { 0 }\).
2. The points \(P\) and \(Q\) have position vectors \(\mathbf { p } = 2 \mathbf { i } + 7 \mathbf { j } - \mathbf { k }\) and \(\mathbf { q } = \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k }\) respectively. Find, in the form \(( \mathbf { r } - \mathbf { a } ) \times \mathbf { d } = \mathbf { 0 }\), the equation of line \(P Q\).

7 The points \(A ( 0,35,120 ) , B ( 28,21,120 )\) and \(C ( 96,35 , - 72 )\) lie on the sphere \(S\), with centre \(O\) and radius 125 . Triangle \(A B C\) is denoted by \(\triangle\).

1. Find, in simplest surd form, the area of \(\Delta\). The points \(A , B\) and \(C\) also form a spherical triangle, \(T\), on the surface of \(S\). Each 'side' of \(T\) is the shorter arc of the circle, centre \(O\) and radius 125, which passes through two of the given vertices of \(T\). In order to find an approximation to the area of the spherical triangle, \(T\) is being modelled by \(\Delta\).
2. (a) State the assumption being made in using this model.
   (b) Say, giving a reason, whether the model gives an under-estimate or an over-estimate of the area of \(T\).

8 In this question you must show detailed reasoning. A sequence of vectors \(\mathbf { a } _ { 1 } , \mathbf { a } _ { 2 } , \mathbf { a } _ { 3 } , \ldots\) is defined by

• \(\mathbf { a } _ { 1 } = \left( \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right)\)
• \(\quad \mathbf { a } _ { n + 1 } = \left( \mathbf { a } _ { n } \times \mathbf { b } \right) \times \mathbf { b }\), for integers \(n \geqslant 1\), where \(\mathbf { b }\) is the vector \(\frac { 1 } { 4 } \left( \begin{array} { c } - 3 \\ 1 \\ 2 \end{array} \right)\).
  1. Prove by induction that \(\mathbf { a } _ { n } = \left( - \frac { 7 } { 8 } \right) ^ { n - 1 } \left( \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right)\). for all integers \(n \geqslant 1\).
  2. Use an algebraic method to find the smallest value of \(n\) such that \(\left| \mathbf { a } _ { n } \right| < 0.001\).

\section*{END OF QUESTION PAPER}

1 Points \(A , B\) and \(C\) have coordinates \(( 0,1 , - 4 ) , ( 1,1 , - 2 )\) and \(( 3,2,5 )\) respectively.

1. Find the vector product \(\overrightarrow { A B } \times \overrightarrow { A C }\).
2. Hence find the equation of the plane \(A B C\) in the form \(a x + b y + c z = d\).

1. The hyperbola \(H\) has equation \(\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 4 } = 1\).

The line \(l _ { 1 }\) is the tangent to \(H\) at the point \(P ( 4 \sec t , 2 \tan t )\).

1. Use calculus to show that an equation of \(l _ { 1 }\) is $$2 y \sin t = x - 4 \cos t$$ The line \(l _ { 2 }\) passes through the origin and is perpendicular to \(l _ { 1 }\).
   The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(Q\).
2. Show that, as \(t\) varies, an equation of the locus of \(Q\) is $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 16 x ^ { 2 } - 4 y ^ { 2 }$$

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 The point A has coordinates \(( 2,5,4 )\) and the line BC has equation $$\mathbf { r } = \left( \begin{array} { c } 8 \\ 25 \\ 43 \end{array} \right) + \lambda \left( \begin{array} { c } 4 \\ 15 \\ 25 \end{array} \right)$$ You are given that \(\mathrm { AB } = \mathrm { AC } = 15\).

1. Show that the coordinates of one of the points B and C are (4, 10, 18). Find the coordinates of the other point. These points are B and C respectively.
2. Find the equation of the plane ABC in cartesian form.
3. Show that the plane containing the line BC and perpendicular to the plane ABC has equation \(5 y - 3 z + 4 = 0\). The point D has coordinates \(( 1,1,3 )\).
4. Show that \(| \overrightarrow { B C } \times \overrightarrow { A D } | = \sqrt { 7667 }\) and hence find the shortest distance between the lines \(B C\) and \(A D\).
5. Find the volume of the tetrahedron ABCD .

11 The lines \(l _ { 1 } , l _ { 2 }\) and \(l _ { 3 }\) are defined as follows. $$\begin{aligned} & l _ { 1 } : \left( \mathbf { r } - \left[ \begin{array} { c } 1 \\ 5 \\ - 1 \end{array} \right] \right) \times \left[ \begin{array} { c } - 2 \\ 1 \\ - 3 \end{array} \right] = \mathbf { 0 } \\ & l _ { 2 } : \left( \mathbf { r } - \left[ \begin{array} { c } - 3 \\ 2 \\ 7 \end{array} \right] \right) \times \left[ \begin{array} { c } 2 \\ - 1 \\ 3 \end{array} \right] = \mathbf { 0 } \\ & l _ { 3 } : \left( \mathbf { r } - \left[ \begin{array} { c } - 5 \\ 12 \\ - 4 \end{array} \right] \right) \times \left[ \begin{array} { l } 4 \\ 0 \\ 9 \end{array} \right] = \mathbf { 0 } \end{aligned}$$ 11

1. 1. Explain how you know that two of the lines are parallel.
      

      11 (ii) Show that the perpendicular distance between these two parallel lines is 7.95 units, correct to three significant figures. [5 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

      11
2. Show that the lines \(l _ { 1 }\) and \(l _ { 3 }\) meet, and find the coordinates of their point of intersection. \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-23_2488_1716_219_153}

3 The position vector of point \(A\) is \(\mathbf { a } = - 9 \mathbf { i } + 2 \mathbf { j } + 6 \mathbf { k }\).
The line \(l\) passes through \(A\) and is perpendicular to \(\mathbf { a }\).

1. Determine the shortest distance between the origin, \(O\), and \(l\). \(l\) is also perpendicular to the vector \(\mathbf { b }\) where \(\mathbf { b } = - 2 \mathbf { i } + \mathbf { j } + \mathbf { k }\).
2. Find a vector which is perpendicular to both \(\mathbf { a }\) and \(\mathbf { b }\).
3. Write down an equation of \(l\) in vector form. \(P\) is a point on \(l\) such that \(P A = 2 O A\).
4. Find angle \(P O A\) giving your answer to 3 significant figures. \(C\) is a point whose position vector, \(\mathbf { c }\), is given by \(\mathbf { c } = p \mathbf { a }\) for some constant \(p\). The line \(m\) passes through \(C\) and has equation \(\mathbf { r } = \mathbf { c } + \mu \mathbf { b }\). The point with position vector \(9 \mathbf { i } + 8 \mathbf { j } - 12 \mathbf { k }\) lies on \(m\).
5. Find the value of \(p\).

9 The points \(P ( 3,5 , - 21 )\) and \(Q ( - 1,3 , - 16 )\) are on the ceiling of a long straight underground tunnel. A ventilation shaft must be dug from the point \(M\) on the ceiling of the tunnel midway between \(P\) and \(Q\) to horizontal ground level (where the \(z\)-coordinate is 0 ). The ventilation shaft must be perpendicular to the tunnel. The path of the ventilation shaft is modelled by the vector equation \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\), where \(\mathbf { a }\) is the position vector of \(M\). You are given that \(\mathbf { b } = \left( \begin{array} { l } 1 \\ \mathrm {~s} \\ \mathrm { t } \end{array} \right)\) where \(s\) and \(t\) are real numbers.

1. Show that \(\mathrm { S } = 2.5 \mathrm { t } - 2\).
2. Show that at the point where the ventilation shaft reaches the ground \(\lambda = \frac { \mathrm { C } } { \mathrm { t } }\), where \(c\) is a constant to be determined.
3. Using the results in parts (a) and (b), determine the shortest possible length of the ventilation shaft.
4. Explain what the fact that \(\mathbf { b } \times \left( \begin{array} { l } 0 \\ 0 \\ 1 \end{array} \right) \neq \mathbf { O }\) means about the direction of the ventilation shaft.

2 The points \(A\) and \(B\) have position vectors \(\mathbf { a }\) and \(\mathbf { b }\) relative to the origin \(O\). It is given that \(\mathbf { a } = \left( \begin{array} { c } 2 \\ 4 \\ 3 \lambda \end{array} \right)\) and \(\mathbf { b } = \left( \begin{array} { r } \lambda \\ - 4 \\ 6 \end{array} \right)\), where \(\lambda\) is a real parameter.

1. In the case when \(\lambda = 3\), determine the area of triangle \(O A B\).
2. Determine the value of \(\lambda\) for which \(\mathbf { a } \times \mathbf { b } = \mathbf { 0 }\).

1 The points \(A , B\) and \(C\) have position vectors \(\mathbf { a } = \left( \begin{array} { l } 3 \\ 0 \\ 0 \end{array} \right) , \mathbf { b } = \left( \begin{array} { l } 0 \\ 4 \\ 0 \end{array} \right)\) and \(\mathbf { c } = \left( \begin{array} { l } 0 \\ 0 \\ 1 \end{array} \right)\) respectively, relative to the origin \(O\).

1. 1. Calculate \(\mathbf { a } \times \mathbf { b }\), giving your answer as a multiple of \(\mathbf { c }\).
   2. Explain, geometrically, why \(\mathbf { a } \times \mathbf { b }\) must be a multiple of \(\mathbf { c }\).
2. Use a vector product method to calculate the area of triangle \(A B C\).

2
The position vector of point \(A\) is \(\mathbf { a } = - 9 \mathbf { i } + 2 \mathbf { j } + 6 \mathbf { k }\).
The line \(l\) passes through \(A\) and is perpendicular to \(\mathbf { a }\).

1. Determine the shortest distance between the origin, \(O\), and \(l\). \(l\) is also perpendicular to the vector \(\mathbf { b }\) where \(\mathbf { b } = - 2 \mathbf { i } + \mathbf { j } + \mathbf { k }\).
2. Find a vector which is perpendicular to both \(\mathbf { a }\) and \(\mathbf { b }\).
3. Write down an equation of \(l\) in vector form. \(P\) is a point on \(l\) such that \(P A = 2 O A\).
4. Find angle \(P O A\) giving your answer to 3 significant figures. \(C\) is a point whose position vector, \(\mathbf { c }\), is given by \(\mathbf { c } = p \mathbf { a }\) for some constant \(p\). The line \(m\) passes through \(C\) and has equation \(\mathbf { r } = \mathbf { c } + \mu \mathbf { b }\). The point with position vector \(9 \mathbf { i } + 8 \mathbf { j } - 12 \mathbf { k }\) lies on \(m\).
5. Find the value of \(p\). \section*{In this question you must show detailed reasoning.} You are given that \(\alpha , \beta\) and \(\gamma\) are the roots of the equation \(5 x ^ { 3 } - 2 x ^ { 2 } + 3 x + 1 = 0\).
   1. Find the value of \(\alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 }\).
   2. Find a cubic equation whose roots are \(\alpha ^ { 2 } , \beta ^ { 2 }\) and \(\gamma ^ { 2 }\) giving your answer in the form \(a x ^ { 3 } + b x ^ { 2 } + c x + d = 0\) where \(a , b , c\) and \(d\) are integers.

3 The equations of two intersecting lines are \(\mathbf { r } = \left( \begin{array} { c } - 12 \\ a \\ - 1 \end{array} \right) + \lambda \left( \begin{array} { l } 2 \\ 2 \\ 1 \end{array} \right) \quad \mathbf { r } = \left( \begin{array} { l } 2 \\ 0 \\ 5 \end{array} \right) + \mu \left( \begin{array} { c } - 3 \\ 1 \\ - 1 \end{array} \right)\) where \(a\) is a constant.

1. Find a vector, \(\mathbf { b }\), which is perpendicular to both lines.
2. Show that b. \(\left( \begin{array} { c } - 12 \\ a \\ - 1 \end{array} \right) =\) b. \(\left( \begin{array} { l } 2 \\ 0 \\ 5 \end{array} \right)\).
3. Hence, or otherwise, find the value of \(a\).

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}

9 Three non-collinear points \(A , B\) and \(C\) have position vectors \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\) respectively, relative to the origin \(O\). The plane through \(A , B\) and \(C\) is denoted by \(\Pi\).

1. (a) Prove that the area of triangle \(A B C\) is \(\frac { 1 } { 2 } | \mathbf { a } \times \mathbf { b } + \mathbf { b } \times \mathbf { c } + \mathbf { c } \times \mathbf { a } |\).
   (b) Describe the significance of the vector \(\mathbf { a } \times \mathbf { b } + \mathbf { b } \times \mathbf { c } + \mathbf { c } \times \mathbf { a }\) in relation to \(\Pi\).
2. (a) In the case when \(\mathbf { a } = a \mathbf { i } , \mathbf { b } = b \mathbf { j }\) and \(\mathbf { c } = c \mathbf { k }\), where \(a , b\) and \(c\) are positive scalar constants, determine the equation of \(\Pi\) in the form r.n \(= d\), where the components of \(\mathbf { n }\) and the value of the scalar constant \(d\) are to be given in terms of \(a , b\) and \(c\).
   (b) Deduce the shortest distance from the origin \(O\) to \(\Pi\) in this case.