Find the Cartesian equation of plane Π containing the points \({\text{A}}\left( {6,{\text{ }}2,{\text{ }}1} \right)\) and \({\text{B}}\left( {3,{\text{ }} - 1,{\text{ }}1} \right)\) and perpendicular to the plane \(x + 2y - z - 6 = 0\).

Given that a \( \times \) b \( = \) b \( \times \) c \( \ne \) 0 prove that a \( + \) c \( = \) sb where s is a scalar.

Find a vector equation for the line, \({L_1}\), which passes through the points P and Q.

The line \({L_2}\) has equation

\[r = \left( {\begin{array}{*{20}{c}}
  { - 4} \\
  0 \\
  4
\end{array}} \right) + s\left( {\begin{array}{*{20}{c}}
  5 \\
  2 \\
  0
\end{array}} \right){\text{.}}\]

Show that \({L_1}\) and \({L_2}\) intersect at the point R(1, 2, 4).

Find the acute angle between \({L_1}\) and \({L_2}\).

Let S be a point on \({L_2}\) such that \(\left| {\overrightarrow {{\text{RP}}} } \right| = \left| {\overrightarrow {{\text{RS}}} } \right|\). 

Show that one of the possible positions for S is \({{\text{S}}_1}\)(−4, 0, 4) and find the coordinates of the other possible position, \({{\text{S}}_2}\).

Let S be a point on \({L_2}\) such that \(\left| {\overrightarrow {{\text{RP}}} } \right| = \left| {\overrightarrow {{\text{RS}}} } \right|\).

Find a vector equation of the line which passes through R and bisects \({\rm{P\hat R}}{{\text{S}}_1}\).

Find the coordinates of S.

The vector product \(\overrightarrow {{\text{PQ}}}  \times \overrightarrow {{\text{PS}}} = \left( {\begin{array}{*{20}{c}}
  { - 13} \\
  7 \\
  m
\end{array}} \right)\). Find the value of m .

Hence calculate the area of parallelogram PQRS.

Find the Cartesian equation of the plane, \({\prod _1}\) , containing the parallelogram PQRS.

Write down the vector equation of the line through the origin (0, 0, 0) that is perpendicular to the plane \({\prod _1}\) .

Hence find the point on the plane that is closest to the origin.

A second plane, \({\prod _2}\) , has equation x − 2y + z = 3. Calculate the angle between the two planes.

The vector equation of line \(l\) is given as \(\left( {\begin{array}{*{20}{c}}
  x \\
  y \\
  z
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
  1 \\
  3 \\
  6
\end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}}
  { - 1} \\
  2 \\
  { - 1}
\end{array}} \right)\) .

Find the Cartesian equation of the plane containing the line \(l\) and the point A(4, − 2, 5) .

Consider the vectors a \( = \sin (2\alpha )\)i \( - \cos (2\alpha )\)j + k and b \( = \cos \alpha \)i \( - \sin \alpha \)j − k, where \(0 < \alpha < 2\pi \).

Let \(\theta \) be the angle between the vectors a and b.

(a)     Express \(\cos \theta \) in terms of \(\alpha \).

(b)     Find the acute angle \(\alpha \) for which the two vectors are perpendicular.

(c)     For \(\alpha = \frac{{7\pi }}{6}\), determine the vector product of a and b and comment on the geometrical significance of this result.

Find the Cartesian equation of \(\pi \).

The line \({l_3}\) passing through the point \((4,{\text{ }}0,{\text{ }}8)\) is perpendicular to \(\pi \).

Find the coordinates of the point where \({l_3}\) meets \(\pi \).

(a)     If \(a = 4\) find the coordinates of the point of intersection of the three planes.

(b)     (i)     Find the value of \(a\) for which the planes do not meet at a unique point.

  (ii)     For this value of \(a\) show that the three planes do not have any common point.

Find \(\overrightarrow {{\text{OA}}}  \times \overrightarrow {{\text{OB}}} \).

Hence find the area of the triangle OAB.

(i)     Find \(\overrightarrow {{\text{OA}}}  \times \overrightarrow {{\text{OB}}} \) .

(ii)     Determine the area of the triangle OAB.

(iii)     Find the Cartesian equation of the plane OAB.

(i)     Find the vector equation of the line \({L_1}\) containing the points A and B.

(ii)     The line \({L_2}\) has vector equation \(\left( {\begin{array}{*{20}{c}}
  x \\
  y \\
  z
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
  2 \\
  4 \\
  3
\end{array}} \right) + \mu \left( {\begin{array}{*{20}{c}}
  1 \\
  3 \\
  2
\end{array}} \right)\).

Determine whether or not \({L_1}\) and \({L_2}\) are skew.

Find the angle between the planes \({\pi _1}\)and \({\pi _2}\) .

The planes \({\pi _1}\) and \({\pi _2}\) intersect in the line \({L_1}\) . Show that the vector equation of

\({L_1}\) is \(r = \left( {\begin{array}{*{20}{c}}
0\\
{2 - 3k}\\
{2k - 2}
\end{array}} \right) + t\left( {\begin{array}{*{20}{c}}
1\\
5\\
{ - 3}
\end{array}} \right)\)

The line \({L_2}\) has Cartesian equation \(5 - x = y + 3 = 2 - 2z\) . The lines \({L_1}\) and \({L_2}\) intersect at a point X. Find the coordinates of X.

Determine a Cartesian equation of the plane \({\pi _3}\) containing both lines \({L_1}\) and \({L_2}\) .

Let Y be a point on \({L_1}\) and Z be a point on \({L_2}\) such that XY is perpendicular to YZ and the area of the triangle XYZ is 3. Find the perimeter of the triangle XYZ.

Find the vector equation of the line of intersection of the three planes represented by the following system of equations.

\[2x - 7y + 5z = 1\]

\[6x + 3y - z = - 1\]

\[ - 14x - 23y + 13z = 5\]

Show that the two submarines would collide at a point P and write down the coordinates of P.

Show that submarine B travels in the same direction as originally planned.

Find the value of t when submarine B passes through P.

Find an expression for the distance between the two submarines in terms of t.

Find the value of t when the two submarines are closest together.

Find the distance between the two submarines at this time.

Find the vector equation of the line (BC).

Determine whether or not the lines (OA) and (BC) intersect.

Find the Cartesian equation of the plane Π\(_1\), which passes through C and is perpendicular to \(\overrightarrow {{\rm{OA}}} \).

Show that the line (BC) lies in the plane Π\(_1\).

Verify that 2j + k is perpendicular to the plane Π\(_2\).

Find a vector perpendicular to the plane Π\(_3\).

Find the acute angle between the planes Π\(_2\) and Π\(_3\).

Show that

(i)     \({\left| {\overrightarrow {{\text{OC}}} } \right|^2} = |\)a\({|^2} + 2\)a \( \bullet \) b \( + |\)b\({|^2}\);

(ii)     \({\left| {\overrightarrow {{\text{AB}}} } \right|^2} = |\)a\({|^2} - 2\)a \( \bullet \) b \( + |\)b\({|^2}\).

Given that \(\left| {\overrightarrow {{\text{OC}}} } \right| = \left| {\overrightarrow {{\text{AB}}} } \right|\), prove that OACB is a rectangle.

find the vectors \({{a}}\) and \({{b}}\).

Roderick is at a point \({\text{C}}(11,{\text{ }}9)\). During Ed’s walk from \(P\) to \(Q\) Roderick wishes to signal to Ed. He decides to signal when Ed is at the closest point to \(C\).

Find the time when Roderick signals to Ed.

Find the values of k for which the following system of equations has no solutions and the value of k for the system to have an infinite number of solutions.

\[x - 3y + z = 3\]

\[x + 5y - 2z = 1\]

\[16y - 6z = k\]

Given that the system of equations can be solved, find the solutions in the form of a vector equation of a line, r = a + λb , where the components of b  are integers.

The plane \( \div \) is parallel to both the line in part (b) and the line \(\frac{{x - 4}}{3} = \frac{{y - 6}}{{ - 2}} = \frac{{z - 2}}{0}\).

Given that \( \div \) contains the point (1, 2, 0) , show that the Cartesian equation of ÷ is 16x + 24y − 11z = 64 .

The z-axis meets the plane \( \div \) at the point P. Find the coordinates of P.

Find the angle between the line \(\frac{{x - 2}}{3} = \frac{{y + 5}}{4} = \frac{z}{2}\) and the plane \( \div \) .

(a)     Find the coordinates of the point \(A\) on \({l_1}\) and the point \(B\) on \({l_2}\) such that \(\overrightarrow {{\text{AB}}} \) is perpendicular to both \({l_1}\) and \({l_2}\) .

(b)     Find \(\left| {{\text{AB}}} \right|\) .

(c)     Find the Cartesian equation of the plane \(\prod \) which contains \({l_1}\) and does not intersect \({l_2}\) .

The angle between the vector a = i − 2j + 3k and the vector b = 3i − 2j + mk is 30° .

Find the values of m.

Find the angle between the lines \(\frac{{x - 1}}{2} = 1 - y = 2z\) and \(x = y = 3z\) .

Show that \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{5y - 2x}}{{2y - 5x}}\).

Find the equation of the normal to the curve at the point \((6,{\text{ }}1)\).

Find the distance between the two points on the curve where each tangent is parallel to the line \(y = x\).

The diagram shows a cube OABCDEFG.

Let O be the origin, (OA) the x-axis, (OC) the y-axis and (OD) the z-axis.

Let M, N and P be the midpoints of [FG], [DG] and [CG], respectively.

The coordinates of F are (2, 2, 2).

(a)     Find the position vectors \(\overrightarrow {{\text{OM}}} \), \(\overrightarrow {{\text{ON}}} \) and \(\overrightarrow {{\text{OP}}} \) in component form.

(b)     Find \(\overrightarrow {{\text{MP}}}  \times \overrightarrow {{\text{MN}}} \).

(c)     Hence,

  (i)     calculate the area of the triangle MNP;

  (ii)     show that the line (AG) is perpendicular to the plane MNP;

  (iii)     find the equation of the plane MNP.

(d)     Determine the coordinates of the point where the line (AG) meets the plane MNP.

Show that OC = 2AB .

Find the position vectors of C, D and E in terms of a and b .

Find the vector equation of L, the line of intersection of \({\Pi _1}\) and \({\Pi _2}\).

Show that the plane \({\Pi _3}\) which is perpendicular to \({\Pi _1}\) and contains L, has equation \(x - 2z = 1\).

The point P has coordinates (−2, 4, 1) , the point Q lies on \({\Pi _3}\) and PQ is perpendicular to \({\Pi _2}\). Find the coordinates of Q.

Consider the two planes

     \({\pi _1}:4x + 2y - z = 8\)

     \({\pi _2}:x + 3y + 3z = 3\).

Find the angle between \({\pi _1}\) and \({\pi _2}\), giving your answer correct to the nearest degree.

A ray of light coming from the point (−1, 3, 2) is travelling in the direction of vector \(\left( {\begin{array}{*{20}{c}}
  4 \\
  1 \\
  { - 2}
\end{array}} \right)\) and meets the plane \(\pi :x + 3y + 2z - 24 = 0\) .

Find the angle that the ray of light makes with the plane.

show that \(3{\sin ^2}\theta  - 7\sin \theta  + 2 = 0\);

find the smallest possible positive value of \(\theta \).

Given that \({\boldsymbol{a}} = 2\sin \theta {\boldsymbol{i}} + \left( {1 - \sin \theta } \right){\boldsymbol{j}}\) , find the value of the acute angle \(\theta \) , so that \(\boldsymbol{a}\) is perpendicular to the line \(x + y = 1\).

A line \({L_1}\) has equation r = \(\left( \begin{array}{c} - 5\\ - 3\\2\end{array} \right) + \lambda \left( \begin{array}{c} - 1\\2\\2\end{array} \right)\).

A line \({L_2}\) passing through the origin intersects \({L_1}\) and is perpendicular to \({L_1}\).

(a)     Find a vector equation of \({L_2}\).

(b)     Determine the shortest distance from the origin to \({L_1}\).

Port A is defined to be the origin of a set of coordinate axes and port B is located at the point (70, 30), where distances are measured in kilometres. A ship S₁ sails from port A at 10:00 in a straight line such that its position \(t\) hours after 10:00 is given by \(r = t\left( {\begin{array}{*{20}{c}}
  {10} \\
  {20}
\end{array}} \right)\).
A speedboat S₂ is capable of three times the speed of S₁ and is to meet S₁ by travelling the shortest possible distance. What is the latest time that S₂ can leave port B?

Show that AB = AC and that \({\rm{B\hat AC}} = 60^\circ \).

Find the Cartesian equation of \(\Pi \), the plane passing through A, B, and C.

 (i)     Find the Cartesian equation of \({\Pi _1}\) , the plane perpendicular to (AB) passing through the midpoint of [AB] .

(ii)     Find the Cartesian equation of \({\Pi _2}\) , the plane perpendicular to (AC) passing through the midpoint of [AC].

Find the vector equation of L , the line of intersection of \({\Pi _1}\) and \({\Pi _2}\) , and show that it is perpendicular to \(\Pi \) .

A methane molecule consists of a carbon atom with four hydrogen atoms symmetrically placed around it in three dimensions.

The positions of the centres of three of the hydrogen atoms are A, B and C as given. The position of the centre of the fourth hydrogen atom is D.

Using the fact that \({\text{AB}} = {\text{AD}}\) , show that the coordinates of one of the possible positions of the fourth hydrogen atom is \(( -1,\,4,\,5)\) .

A methane molecule consists of a carbon atom with four hydrogen atoms symmetrically placed around it in three dimensions.

The positions of the centres of three of the hydrogen atoms are A, B and C as given. The position of the centre of the fourth hydrogen atom is D.

Letting D be \(( - 1,\,4,\,5)\) , show that the coordinates of G, the position of the centre of the carbon atom, are \((2,\,1,\,2)\) . Hence calculate \({\rm{D}}\hat {\rm{G}}{\rm{A}}\) , the bonding angle of carbon.

Show that the lines \({L_1}\) and \({L_2}\) are skew.

Find the acute angle between the lines \({L_1}\) and \({L_2}\).

(i)     Find a vector perpendicular to both lines.

(ii)     Hence determine an equation of the line \({L_3}\) that is perpendicular to both \({L_1}\) and \({L_2}\) and intersects both lines.

Find the acute angle between the planes with equations \(x + y + z = 3\) and \(2x - z = 2\).

The planes \(2x + 3y - z = 5\) and \(x - y + 2z = k\) intersect in the line \(5x + 1 = 9 - 5y = - 5z\) .

Find the value of k .

(a)     Write the vector equations of the following lines in parametric form.

\[{r_1} = \left( {\begin{array}{*{20}{c}}
  3 \\
  2 \\
  7
\end{array}} \right) + m\left( {\begin{array}{*{20}{c}}
  2 \\
  { - 1} \\
  2
\end{array}} \right)\]

\[{r_2} = \left( {\begin{array}{*{20}{c}}
  1 \\
  4 \\
  2
\end{array}} \right) + n\left( {\begin{array}{*{20}{c}}
  4 \\
  { - 1} \\
  1
\end{array}} \right)\]

(b)     Hence show that these two lines intersect and find the point of intersection, A.

(c)     Find the Cartesian equation of the plane \(\prod \) that contains these two lines.

(d)     Let B be the point of intersection of the plane \(\prod \) and the line\({r} = \left( {\begin{array}{*{20}{c}}
  { - 8} \\
  { - 3} \\
  0
\end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}}
  3 \\
  8 \\
  2
\end{array}} \right)\).

Find the coordinates of B.

(e)     If C is the mid-point of AB, find the vector equation of the line perpendicular to the plane \(\prod \) and passing through C.

A plane \(\pi \) has vector equation r = (−2i + 3j − 2k) + \(\lambda \)(2i + 3j + 2k) + \(\mu \)(6i − 3j + 2k).

(a)     Show that the Cartesian equation of the plane \(\pi \) is 3x + 2y − 6z = 12.

(b)     The plane \(\pi \) meets the x, y and z axes at A, B and C respectively. Find the coordinates of A, B and C.

(c)     Find the volume of the pyramid OABC.

(d)     Find the angle between the plane \(\pi \) and the x-axis.

(e)     Hence, or otherwise, find the distance from the origin to the plane \(\pi \).

(f)     Using your answers from (c) and (e), find the area of the triangle ABC.

Obtain an expression for \(f'(x)\) .

Sketch the graphs of f and \(f'\) on the same axes, showing clearly all x-intercepts.

Find the x-coordinates of the two points of inflexion on the graph of f .

Find the equation of the normal to the graph of f where x = 0.75 , giving your answer in the form y = mx + c .

Consider the points \({\text{A}}\left( {a{\text{ }},{\text{ }}f(a)} \right)\) , \({\text{B}}\left( {b{\text{ }},{\text{ }}f(b)} \right)\) and \({\text{C}}\left( {c{\text{ }},{\text{ }}f(c)} \right)\) where a , b and c \((a < b < c)\) are the solutions of the equation \(f(x) = f'(x)\) . Find the area of the triangle ABC.

(a)     Find the coordinates of P when \(t = 0\) .

(b)     Show that P moves along the line \(L\) with Cartesian equations\[x - 1 = \frac{{y - 2}}{{ - 2}} = \frac{{z + 1}}{3}\]

(c)     (i)     Find the value of t when P lies on the plane with equation \(2x + y + z = 6\) .

  (ii)     State the coordinates of P at this time.

  (iii)     Hence find the total distance travelled by P before it meets the plane.

The position vector at time \(t\) of another point, Q, is given by\[\overrightarrow {{\text{OQ}}}  = \left( {\begin{array}{*{20}{c}}
  {{t^2}} \\
  {1 - t} \\
  {1 - {t^2}}
\end{array}} \right),{\text{ }}t \geqslant 0.\]

(d)     (i)     Find the value of t for which the distance from Q to the origin is minimum.

  (ii)     Find the coordinates of Q at this time.

(e)     Let \(\boldsymbol{a}\) , \(\boldsymbol{b}\) and \(\boldsymbol{c}\) be the position vectors of Q at times \(t = 0\), \(t =1\) and \(t = 2\) respectively.

  (i)     Show that the equation \({\boldsymbol{a}} - {\boldsymbol{b}} = k\left( {{\boldsymbol{b}} - {\boldsymbol{c}}} \right)\) has no solution for \(k\) .

  (ii)     Hence show that the path of Q is not a straight line.