Vector Product and Surfaces

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 \(B\).

1 A surface has equation \(z = x \tan y\) for \(- \frac { 1 } { 2 } \pi < y < \frac { 1 } { 2 } \pi\).

1. Find
   • \(\frac { \partial z } { \partial x }\),
   • \(\frac { \partial z } { \partial y }\).
   • Find in cartesian form, the equation of the tangent plane to the surface at the point where \(x = 1\) and \(y = \frac { 1 } { 4 } \pi\).

14 Given that the vectors \(\mathbf { a }\) and \(\mathbf { b }\) are perpendicular, prove that \(| ( \mathbf { a } + 5 \mathbf { b } ) \times ( \mathbf { a } - 4 \mathbf { b } ) | = k | \mathbf { a } | | \mathbf { b } |\), where \(k\) is an integer to be found. Explicitly state any properties of the vector product that you use within your proof.
[0pt] [9 marks] LL

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

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 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. \(y = 1\)

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.

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 \(Q ( 1,2,38 )\).

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.

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 \(( \mathbf { r } - \mathbf { a } ) \times \mathbf { d } = \mathbf { 0 }\).

5 A surface \(S\) is defined for \(z \geqslant 0\) by \(x ^ { 2 } + y ^ { 2 } + 2 z ^ { 2 } = 126\). \(C\) is the set of points on \(S\) for which the tangent plane to \(S\) at that point intersects the \(x - y\) plane at an angle of \(\frac { 1 } { 3 } \pi\) radians. Show that \(C\) lies in a plane, \(\Pi\), whose equation should be determined.

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 \(\frac { 1 } { 2 } \sqrt { 3 } ( 1 + \sqrt { 51 } )\).

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 \(A B C D\).

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}

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 \(k\).

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 \(\mathbf { a } \cdot ( \mathbf { c } \times \mathbf { d } )\).

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 \(| O D | = \frac { | \mathbf { a } \times \mathbf { b } | } { | \mathbf { b } - \mathbf { a } | }\).

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 \(a x + b y + c z = d\).
      [0pt] [BLANK PAGE]
   3. The equation of the curve shown on the graph is, in polar coordinates, \(r = 3 \sin 2 \theta\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). \includegraphics[max width=\textwidth, alt={}, center]{25055c9c-2d29-476e-887a-a10699814b85-04_505_704_348_292}
   4. The greatest value of \(r\) on the curve occurs at the point \(P\).
      1. Show that \(\theta = \frac { 1 } { 4 } \pi\) at the point \(P\).
      2. Find the value of \(r\) at the point \(P\).
      3. Mark the point \(P\) on a copy of the graph.
   5. In this question you must show detailed reasoning.
   Find the exact area of the region enclosed by the curve.
   [0pt] [BLANK PAGE]