Vector Product and Surfaces

A transformation is equivalent to a shear parallel to the \(x\)-axis followed by a shear parallel to the \(y\)-axis and is represented by the matrix \(\begin{pmatrix} 1 & s \\ t & 0 \end{pmatrix}\). Find in terms of \(s\) the matrices which represent each of the shears. [7]

1 The points \(A , B\) and \(C\) have position vectors \(6 \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k } , 13 \mathbf { i } + 2 \mathbf { j } + 5 \mathbf { k }\) and \(16 \mathbf { i } + 6 \mathbf { j } + 3 \mathbf { k }\) respectively.

1. Using the vector product, calculate the area of triangle \(A B C\).
2. Hence find, in simplest surd form, the perpendicular distance from \(C\) to the line through \(A\) and \(B\).

2 Find the volume of tetrahedron OABC , where O is the origin, \(\mathrm { A } = ( 2,3,1 ) , \mathrm { B } = ( - 4,2,5 )\) and \(\mathrm { C } = ( 1,4,4 )\).

5 The surface \(S\) has equation \(x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = x y z - 1\).

1. Show that \(( 2 z - x y ) \left( x \frac { \partial z } { \partial x } + y \frac { \partial z } { \partial y } \right) = 2 \left( 1 + z ^ { 2 } \right)\).
2. Deduce that \(S\) has no stationary point.

6 A surface \(S\) has equation \(z = \mathrm { f } ( x , y )\), where \(\mathrm { f } ( x , y ) = 2 x ^ { 2 } - y ^ { 2 } + 3 x y + 17 y\). It is given that \(S\) has a single stationary point, \(P\).

1. (a) Determine the coordinates of \(P\).
   (b) Determine the nature of \(P\).
2. Find the equation of the tangent plane to \(S\) at the point \(Q ( 1,2,38 )\).

2 A surface \(S\) has equation \(\mathrm { z } = 4 \mathrm { x } \sqrt { \mathrm { y } } - \mathrm { y } \sqrt { \mathrm { x } } + \mathrm { y } ^ { 2 }\) for \(x , y \geqslant 0\). Determine the equation of the tangent plane to \(S\) at the point (1,4,20). Give your answer in the form \(\mathrm { ax } + \mathrm { by } + \mathrm { cz } = \mathrm { d }\) where \(a , b , c\) and \(d\) are integers.

It is given that $$\begin{bmatrix} 2 \\ 1 \\ 9 \end{bmatrix} \times \begin{bmatrix} 5 \\ \lambda \\ -6 \end{bmatrix} = 0$$ where \(\lambda\) is a constant. Find the value of \(\lambda\) Circle your answer. [1 mark] \(-28\) \quad\quad \(-8\) \quad\quad \(8\) \quad\quad \(28\)

3 The non-zero vectors \(\mathbf { x }\) and \(\mathbf { y }\) are such that \(\mathbf { x } \times \mathbf { y } = \mathbf { 0 }\).

1. Explain the geometrical significance of this statement.
2. Use your answer to part (a) to explain how the line equation \(\mathbf { r } = \mathbf { a } + t \mathbf { d }\) can be written in the form \(( \mathbf { r } - \mathbf { a } ) \times \mathbf { d } = \mathbf { 0 }\).

2 A surface has equation \(x ^ { 2 } - 4 x y + 3 y ^ { 2 } - 2 z ^ { 2 } - 63 = 0\).

1. Find a normal vector at the point \(( x , y , z )\) on the surface.
2. Find the equation of the tangent plane to the surface at the point \(\mathrm { Q } ( 17,4,1 )\).
3. The point \(( 17 + h , 4 + p , 1 - h )\), where \(h\) and \(p\) are small, is on the surface and is close to Q . Find an approximate expression for \(p\) in terms of \(h\).
4. Show that there is no point on the surface where the normal line is parallel to the \(z\)-axis.
5. Find the two values of \(k\) for which \(5 x - 6 y + 2 z = k\) is a tangent plane to the surface.

1 The surface \(S\) is defined for all real \(x\) and \(y\) by the equation \(z = x ^ { 2 } + 2 x y\). The intersection of \(S\) with the plane \(\Pi\) gives a section of the surface. On the axes provided in the Printed Answer Booklet, sketch this section when the equation of \(\Pi\) is each of the following.

1. \(x = 1\)
2. \(y = 1\)

1. Find, in terms of \(x\), a vector which is perpendicular to the vectors \(\begin{pmatrix} x-2 \\ 5 \\ 1 \end{pmatrix}\) and \(\begin{pmatrix} x \\ 6 \\ 2 \end{pmatrix}\). [2]
2. Find the shortest possible vector of the form \(\begin{pmatrix} 1 \\ a \\ b \end{pmatrix}\) which is perpendicular to the vectors \(\begin{pmatrix} x-2 \\ 5 \\ 1 \end{pmatrix}\) and \(\begin{pmatrix} x \\ 6 \\ 2 \end{pmatrix}\). [5]

1. Vector \(\mathbf{v}\) is perpendicular to both \(\begin{pmatrix} -1 \\ 1 \\ 1 \end{pmatrix}\) and \(\begin{pmatrix} 1 \\ p \\ p^2 \end{pmatrix}\) where \(p\) is a real number. Show that it is impossible for \(\mathbf{v}\) to be perpendicular to the vector \(\begin{pmatrix} 1 \\ 1 \\ p-1 \end{pmatrix}\). [6]

2 The points \(A ( 1,2,2 ) , B ( 8,2,5 ) , C ( - 3,6,5 )\) and \(D ( - 10,6,2 )\) are the vertices of parallelogram \(A B C D\). Determine the area of \(A B C D\).

2 You are given \(\mathrm { g } ( x , y , z ) = 6 x z - ( x + 2 y + 3 z ) ^ { 2 }\).

1. Find \(\frac { \partial \mathrm { g } } { \partial x } , \frac { \partial \mathrm {~g} } { \partial y }\) and \(\frac { \partial \mathrm { g } } { \partial z }\). A surface \(S\) has equation \(\mathrm { g } ( x , y , z ) = 125\).
2. Find the equation of the normal line to \(S\) at the point \(\mathrm { P } ( 7 , - 7.5,3 )\).
3. The point Q is on this normal line and is close to P . At \(\mathrm { Q } , \mathrm { g } ( x , y , z ) = 125 + h\), where \(h\) is small. Find the vector \(\mathbf { n }\) such that \(\overrightarrow { \mathrm { PQ } } = h \mathbf { n }\) approximately.
4. Show that there is no point on \(S\) at which the normal line is parallel to the \(z\)-axis.
5. Find the two points on \(S\) at which the tangent plane is parallel to \(x + 5 y = 0\).

5 The tetrahedron \(T\), shown below, has vertices at \(O ( 0,0,0 ) , A ( 1,2,2 ) , B ( 2,1,2 )\) and \(C ( 2,2,1 )\). \includegraphics[max width=\textwidth, alt={}, center]{59fa1650-a296-471e-93b9-0988177cd89d-3_360_464_319_555} Diagram not drawn to scale Show that the surface area of \(T\) is \(\frac { 1 } { 2 } \sqrt { 3 } ( 1 + \sqrt { 51 } )\).

5 The points \(A\) and \(B\) have position vectors \(\mathbf { a }\) and \(\mathbf { b }\) respectively, relative to a fixed origin \(O\).

1. (a) Prove that \(\mathbf { a } \times ( \mathbf { b } - \mathbf { a } ) = \mathbf { a } \times \mathbf { b }\).
   (b) Determine the relationship between \(\mathbf { a } \times ( \mathbf { b } - \mathbf { a } )\) and \(\mathbf { b } \times ( \mathbf { b } - \mathbf { a } )\).
2. The point \(D\) is on the line \(A B\). \(O D\) is perpendicular to \(A B\). By considering the area of triangle \(O A B\), show
   that \(| O D | = \frac { | \mathbf { a } \times \mathbf { b } | } { | \mathbf { b } - \mathbf { a } | }\).

4 The equation of line \(l\) can be written in either of the following vector forms.

• \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\), where \(\lambda \in \mathbb { R }\)
• \(( \mathbf { r } - \mathbf { c } ) \times \mathbf { d } = \mathbf { 0 }\)
  1. Write down two equations involving the vectors \(\mathbf { a , b , c }\), and d, giving reasons for your answers.
  2. Determine the value of \(\mathbf { a } \cdot ( \mathbf { c } \times \mathbf { d } )\).

2 A surface has equation \(z = x y ^ { 2 } - 4 x ^ { 2 } y - 2 x ^ { 3 } + 27 x ^ { 2 } - 36 x + 20\).

1. Find \(\frac { \partial z } { \partial x }\) and \(\frac { \partial z } { \partial y }\).
2. Find the coordinates of the four stationary points on the surface, showing that one of them is \(( 2,4,8 )\).
3. Sketch, on separate diagrams, the sections of the surface defined by \(x = 2\) and by \(y = 4\). Indicate the point \(( 2,4,8 )\) on these sections, and deduce that it is neither a maximum nor a minimum.
4. Show that there are just two points on the surface where the normal line is parallel to the vector \(36 \mathbf { i } + \mathbf { k }\), and find the coordinates of these points.

2 A surface \(S\) has equation \(z = 8 y ^ { 3 } - 6 x ^ { 2 } y - 15 x ^ { 2 } + 36 x\).

1. Sketch the section of \(S\) given by \(y = - 3\), and sketch the section of \(S\) given by \(x = - 6\). Your sketches should include the coordinates of any stationary points but need not include the coordinates of the points where the sections cross the axes.
2. From your sketches in part (i), deduce that \(( - 6 , - 3 , - 324 )\) is a stationary point on \(S\), and state the nature of this stationary point.
3. Find \(\frac { \partial z } { \partial x }\) and \(\frac { \partial z } { \partial y }\), and hence find the coordinates of the other three stationary points on \(S\).
4. Show that there are exactly two values of \(k\) for which the plane with equation $$120 x - z = k$$ is a tangent plane to \(S\), and find these values of \(k\).

8 The motion of two remote controlled helicopters \(P\) and \(Q\) is modelled as two points moving along straight lines. Helicopter \(P\) moves on the line \(\mathbf { r } = \left( \begin{array} { r } 2 + 4 p \\ - 3 + p \\ 1 + 3 p \end{array} \right)\) and helicopter \(Q\) moves on the line \(\mathbf { r } = \left( \begin{array} { l } 5 + 8 q \\ 2 + q \\ 5 + 4 q \end{array} \right)\).
The function \(z\) denotes \(( P Q ) ^ { 2 }\), the square of the distance between \(P\) and \(Q\).

1. Show that \(z = 26 p ^ { 2 } + 81 q ^ { 2 } - 90 p q - 58 p + 90 q + 50\).
2. Use partial differentiation to find the values of \(p\) and \(q\) for which \(z\) has a stationary point.
3. With the aid of a diagram, explain why this stationary point must be a minimum point, rather than a maximum point or a saddle point.
4. Hence find the shortest possible distance between the two helicopters. The model is now refined by modelling each helicopter as a sphere of radius 0.5 units.
5. Explain how this will change your answer to part (d). \section*{END OF QUESTION PAPER}