A point \(P\), relative to an origin \(O\), has position vector \(\overrightarrow {{\text{OP}}} = \left( {\begin{array}{*{20}{c}} {1 + s} \\ {3 + 2s} \\ {1 - s} \end{array}} \right),{\text{ }}s \in \mathbb{R}\).

Find the minimum length of \(\overrightarrow {{\text{OP}}} \).

Three distinct non-zero vectors are given by \(\overrightarrow {{\text{OA}}} \) = a, \(\overrightarrow {{\text{OB}}} \) = b, and \(\overrightarrow {{\text{OC}}} \) = c .

If \(\overrightarrow {{\text{OA}}} \) is perpendicular to \(\overrightarrow {{\text{BC}}} \) and \(\overrightarrow {{\text{OB}}} \) is perpendicular to \(\overrightarrow {{\text{CA}}} \) , show that \(\overrightarrow {{\text{OC}}} \) is perpendicular to \(\overrightarrow {{\text{AB}}} \).

Consider the vectors \(\overrightarrow {{\text{OA}}} \) = a, \(\overrightarrow {{\text{OB}}} \) = b and \(\overrightarrow {{\text{OC}}} \) = a + b. Show that if \(|\)a\(|\) = \(|\)b\(|\) then (a + b)\( \cdot \)(a − b) = 0. Comment on what this tells us about the parallelogram OACB.

Consider the plane with equation \(4x - 2y - z = 1\) and the line given by the parametric equations

     \(x = 3 - 2\lambda \)

     \(y = (2k - 1) + \lambda \)

     \(z = - 1 + k\lambda .\)

Given that the line is perpendicular to the plane, find

(a)     the value of k;

(b)     the coordinates of the point of intersection of the line and the plane.

Find the coordinates of the common point of the paths of the two boats.

Show that the boats do not collide.

O, A, B and C are distinct points such that \(\overrightarrow {{\text{OA}}}  = \) a, \(\overrightarrow {{\text{OB}}}  = \) b and \(\overrightarrow {{\text{OC}}}  = \) c.

It is given that c is perpendicular to \(\overrightarrow {{\text{AB}}} \) and b is perpendicular to \(\overrightarrow {{\text{AC}}} \).

Prove that a is perpendicular to \(\overrightarrow {{\text{BC}}} \).

Explain why ABCD is a parallelogram.

Using vector algebra, show that \(\mathop {{\text{AD}}}\limits^ \to   = \mathop {{\text{BC}}}\limits^ \to  \).

Show that p = 1, q = 1 and r = 4.

Find the area of the parallelogram ABCD.

Find the vector equation of the straight line passing through M and normal to the plane \(\Pi \) containing ABCD.

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

Find the coordinates of X, Y and Z.

Find YZ.

Show that the points \({\text{O}}(0,{\text{ }}0,{\text{ }}0)\), \({\text{ A}}(6,{\text{ }}0,{\text{ }}0)\), \({\text{B}}({6,{\text{ }}- \sqrt {24} ,{\text{ }}\sqrt {12} })\), \({\text{C}}({0,{\text{ }}- \sqrt {24} ,{\text{ }}\sqrt {12}})\) form a square.

Find the coordinates of M, the mid-point of [OB].

Show that an equation of the plane \({\mathit{\Pi }}\), containing the square OABC, is \(y + \sqrt 2 z = 0\).

Find a vector equation of the line \(L\), through M, perpendicular to the plane \({\mathit{\Pi }}\).

Find the coordinates of D, the point of intersection of the line \(L\) with the plane whose equation is \(y = 0\).

Find the coordinates of E, the reflection of the point D in the plane \({\mathit{\Pi }}\).

(i)     Find the angle \({\rm{O\hat DA}}\).

(ii)     State what this tells you about the solid OABCDE.

(a)     Show that the two planes

\[{\pi _1}:x + 2y - z = 1\]

\[{\pi _2}:x + z =  - 2\]

are perpendicular.

(b)     Find the equation of the plane \({\pi _3}\) that passes through the origin and is perpendicular to both \({\pi _1}\) and \({\pi _2}\).

The following system of equations represents three planes in space.

\[x + 3y + z =  - 1\]

\[x + 2y - 2z = 15\]

\[2x + y - z = 6\]

Find the coordinates of the point of intersection of the three planes.

(i)     Express \(\overrightarrow {{\text{AM}}} \) in terms of \(a\) and \(c\).

(ii)     Hence show that \(\overrightarrow {{\text{BM}}}  = \frac{1}{2}\)\(a\) – \(b\)\( + \frac{1}{2}c\).

(i)     Express \(\overrightarrow {{\text{RA}}} \) in terms of \(a\) and \(b\).

(ii)     Show that \(\overrightarrow {RT}  =  - \frac{2}{9}a - \frac{2}{9}b + \frac{4}{9}c\).

Prove that \(T\) lies on [\(BM\)].

Find the values of x for which the vectors \(\left( {\begin{array}{*{20}{c}}
  1 \\
  {2\cos x} \\
  0
\end{array}} \right)\) and \(\left( {\begin{array}{*{20}{c}}
  { - 1} \\
  {2\sin x} \\
  1
\end{array}} \right)\) are perpendicular,  \(0 \leqslant x \leqslant \frac{\pi }{2}\).

Find a \( \times \) b.

Hence find the Cartesian equation of the plane containing the vectors a and b, and passing through the point \((1,{\text{ }}0,{\text{ }} - 1)\).

Find the vectors \(\overrightarrow {{\text{AB}}} \) and \(\overrightarrow {{\text{AC}}} \).

Find the Cartesian equation of the plane \(\prod \) that contains the face ABC.

Find the vector \(\overrightarrow {{\text{CA}}}  \times \overrightarrow {{\text{CB}}} \).

Find an exact value for the area of the triangle ABC.

Show that the Cartesian equation of the plane \({\Pi _1}\), containing the triangle ABC, is \(2x + 3y - 4z = 2\).

A second plane \({\Pi _2}\) is defined by the Cartesian equation \({\Pi _2}:4x - y - z = 4\). \({L_1}\) is the line of intersection of the planes \({\Pi _1}\) and \({\Pi _2}\).

Find a vector equation for \({L_1}\).

Find the value of \(\alpha \) if all three planes contain \({L_1}\).

Find conditions on \(\alpha \) and \(\beta \) if the plane \({\Pi _3}\) does not intersect with \({L_1}\).

Find \(\overrightarrow {{\text{BR}}} \) in terms of \({{a}}\), \({{b}}\) and \({{c}}\).

(i)     Find a vector equation of the line that passes through \(B\) and \(R\) in terms of \({{a}}\), \({{b}}\) and \({{c}}\) and a parameter \(\lambda \).

(ii)     Find a vector equation of the line that passes through \(A\) and \(Q\) in terms of \({{a}}\), \({{b}}\) and \({{c}}\) and a parameter \(\mu \).

(iii)     Hence show that \(\overrightarrow {{\text{OG}}}  = \frac{1}{3}({{a}} + {{b}} + {{c}})\) given that \(G\) is the point where [\(BR\)] and [\(AQ\)] intersect.

Show that the line segment [\(CP\)] also includes the point \(G\).

The coordinates of the points \(A\), \(B\) and \(C\) are \((1,{\text{ }}3,{\text{ }}1)\), \((3,{\text{ }}7,{\text{ }} - 5)\) and \((2,{\text{ }}2,{\text{ }}1)\) respectively.

A point \(X\) is such that [\(GX\)] is perpendicular to the plane \(ABC\).

Given that the tetrahedron \(ABCX\) has volume \({\text{12 unit}}{{\text{s}}^{\text{3}}}\), find possible coordinates

of \(X\).

find the value of \(a\);

determine the coordinates of the point of intersection P.

Given the points A(1, 0, 4), B(2, 3, −1) and C(0, 1, − 2) , find the vector equation of the line \({L_1}\) passing through the points A and B.

The line \({L_2}\) has Cartesian equation \(\frac{{x - 1}}{3} = \frac{{y + 2}}{1} = \frac{{z - 1}}{{ - 2}}\).

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

Find the Cartesian equation of the plane \({\Pi _1}\).

(i)     Find the value of \(k\).

(ii)     Find the point of intersection P of the line \({L_3}\) and the plane \({\mathit{\Pi} _2}\).

Given any two non-zero vectors a and b , show that \(|\)a \( \times \) b\({|^2}\) = \(|\)a\({|^2}\)\(|\)b\({|^2}\) – (a \( \cdot \) b)\(^2\).

Find a vector equation of the line L passing through the points A and B.

Find the coordinates of the point of intersection of the line L with the plane Π.

The three vectors \(\boldsymbol{a}\), \(\boldsymbol{b}\) and \(\boldsymbol{c}\) are given by\[{\boldsymbol{a}} = \left( {\begin{array}{*{20}{c}}
  {2y} \\
  { - 3x} \\
  {2x}
\end{array}} \right),{\text{ }}{\boldsymbol{b}}{\text{ }} = \left( {\begin{array}{*{20}{c}}
  {4x} \\
  y \\
  {3 - x}
\end{array}} \right),{\text{ }}{\boldsymbol{c}}{\text{ }} = \left( {\begin{array}{*{20}{c}}
  4 \\
  { - 7} \\
  6
\end{array}} \right){\text{ where }}x,y \in \mathbb{R}{\text{ }}{\text{.}}\]

(a)     If a + 2b − c = 0, find the value of x and of y.

(b)     Find the exact value of \(|\)a + 2b\(|\).

Write down expressions for \(\overrightarrow {{\text{AB}}} \) and \(\overrightarrow {{\text{CB}}} \) in terms of the vectors \({\boldsymbol{a}}\) and \({\boldsymbol{b}}\) .

Hence prove that angle \({\text{A}}\hat {\rm{B}}{\text{C}}\) is a right angle.

Find the cosine of the angle between the two planes in the form \(\sqrt {\frac{p}{q}} \) where \(p,{\text{ }}q \in \mathbb{Z}\).

(i)     Show that \(L\) has direction \(\left( {\begin{array}{*{20}{c}} { - 1} \\ 2 \\ 2 \end{array}} \right)\).

(ii)     Show that the point \({\text{A }}(1,{\text{ }}0,{\text{ }}4)\) lies on both planes.

(iii)     Write down a vector equation of \(L\).

Given the vector \(\overrightarrow {{\text{AB}}} \) is perpendicular to \(L\) find the value of \(a\) and the value of \(b\).

Show that \({\text{AB}} = 3\sqrt 2 \).

Find the coordinates of the two possible positions of \(P\).

Find, in terms of a and b \(\overrightarrow {{\text{OF}}} \).

Find, in terms of a and b \(\overrightarrow {{\text{AF}}} \).

Find an expression for \(\overrightarrow {{\text{OD}}} \) in terms of a, b and \(\lambda \);

Find an expression for \(\overrightarrow {{\text{OD}}} \) in terms of a, b and \(\mu \).

Show that \(\mu  = \frac{1}{{13}}\), and find the value of \(\lambda \).

Deduce an expression for \(\overrightarrow {{\text{CD}}} \) in terms of a and b only.

Given that area \(\Delta {\text{OAB}} = k({\text{area }}\Delta {\text{CAD}})\), find the value of \(k\).

(i)     Find the lengths of the sides of the triangle.

(ii)     Find \(\cos {\rm{B\hat AC}}\).

(i)     Show that \(\overrightarrow {{\text{BC}}}  \times \overrightarrow {{\text{CA}}}  = \) −7i − 3j − 16k.

(ii)     Hence, show that the area of the triangle ABC is \(\frac{1}{2}\sqrt {314} \).

Find the Cartesian equation of the plane containing the triangle ABC.

Find a vector equation of (AB).

The point D on (AB) is such that \(\overrightarrow {{\text{OD}}} \) is perpendicular to \(\overrightarrow {{\text{BC}}} \) where O is the origin.

(i)     Find the coordinates of D.

(ii)     Show that D does not lie between A and B.

Show that \(L\) is not perpendicular to \(\Pi \).

Given that \(L\) lies in the plane \(\Pi \), find the value of \(p\) and the value of \(q\).

(i)     Show that \(p =  - 2\).

(ii)     If \(L\) intersects \(\Pi \) at \(z =  - 1\), find the value of \(q\).

For non-zero vectors \({\boldsymbol{a}}\) and \({\boldsymbol{b}}\), show that

(i)     if \(\left| {{\boldsymbol{a}} - {\boldsymbol{b}}} \right| = \left| {{\boldsymbol{a}} + {\boldsymbol{b}}} \right|\), then \({\boldsymbol{a}}\) and \({\boldsymbol{b}}\) are perpendicular;

(ii)     \({\left| {{\boldsymbol{a}} \times {\boldsymbol{b}}} \right|^2} = {\left| {\boldsymbol{a}} \right|^2}{\left| {\boldsymbol{b}} \right|^2} - {({\boldsymbol{a}} \cdot {\boldsymbol{b}})^2}\).

The points A, B and C have position vectors \({\boldsymbol{a}}\), \({\boldsymbol{b}}\) and \({\boldsymbol{c}}\).

(i)     Show that the area of triangle ABC is \(\frac{1}{2}\left| {{\boldsymbol{a}} \times {\boldsymbol{b}} + {\boldsymbol{b}} \times {\boldsymbol{c}} + {\boldsymbol{c}} \times {\boldsymbol{a}}} \right|\).

(ii)     Hence, show that the shortest distance from B to AC is

\[\frac{{\left| {{\boldsymbol{a}} \times {\boldsymbol{b}} + {\boldsymbol{b}} \times {\boldsymbol{c}} + {\boldsymbol{c}} \times {\boldsymbol{a}}} \right|}}{{\left| {{\boldsymbol{c}} - {\boldsymbol{a}}} \right|}}{\text{.}}\]

Consider the points A(1, −1, 4), B (2, − 2, 5) and O(0, 0, 0).

(a)     Calculate the cosine of the angle between \(\overrightarrow {{\text{OA}}} \) and \(\overrightarrow {{\text{AB}}} \).

(b)     Find a vector equation of the line \({L_1}\) which passes through A and B.

The line \({L_2}\) has equation r = 2i + 4j + 7k + t(2i + j + 3k), where \(t \in \mathbb{R}\) .

(c)     Show that the lines \({L_1}\) and \({L_2}\) intersect and find the coordinates of their point of intersection.

(d)     Find the Cartesian equation of the plane which contains both the line \({L_2}\) and the point A.

Find the area of the parallelogram ABCD.

By using a suitable scalar product of two vectors, determine whether \({\rm{A\hat BC}}\) is acute or obtuse.

The points A, B, C have position vectors i + j + 2k , i + 2j + 3k , 3i + k respectively and lie in the plane \(\pi \) .

(a)     Find

(i)     the area of the triangle ABC;

(ii)     the shortest distance from C to the line AB;

(iii)     the cartesian equation of the plane \(\pi \) .

The line L passes through the origin and is normal to the plane \(\pi \) , it intersects \(\pi \) at the

point D.

(b)     Find

(i)     the coordinates of the point D;

(ii)     the distance of \(\pi \) from the origin.

Find \(\overrightarrow {{\text{AB}}}  \times \overrightarrow {{\text{AC}}} \).

Hence find the area of the triangle ABC.

Find an expression for \(\overrightarrow {{\text{CB}}} \) and for \(\overrightarrow {{\text{AC}}} \) in terms of \(\boldsymbol{b}\) and \(\boldsymbol{c}\) .

Hence prove that \({\rm{A\hat CB}}\) is a right angle.

Find the coordinates of the point of intersection of the planes defined by the equations \(x + y + z = 3,{\text{ }}x - y + z = 5\) and \(x + y + 2z = 6\).

Find the Cartesian equation of the plane \({\Pi _1}\), passing through the points A , B and D.

Find the angle between the faces ABD and BCD.

Find the Cartesian equation of \({\Pi _3}\).

Show that P is the midpoint of AD.

Find the area of the triangle OPQ.

PQRS is a rhombus. Given that \(\overrightarrow {{\text{PQ}}}  = \) \(\boldsymbol{a}\) and \(\overrightarrow {{\text{QR}}}  = \) \(\boldsymbol{b}\),

(a)     express the vectors \(\overrightarrow {{\text{PR}}} \) and \(\overrightarrow {{\text{QS}}} \) in terms of \(\boldsymbol{a}\) and \(\boldsymbol{b}\);

(b)     hence show that the diagonals in a rhombus intersect at right angles.

The vectors a , b , c satisfy the equation a + b + c = 0 . Show that a \( \times \) b = b \( \times \) c = c \( \times \) a .

Consider the vectors a = 6i + 3j + 2k, b = −3j + 4k.

(i)     Find the cosine of the angle between vectors a and b.

(ii)     Find a \( \times \) b.

(iii)     Hence find the Cartesian equation of the plane \(\prod \) containing the vectors a and b and passing through the point (1, 1, −1).

(iv)     The plane \(\prod \) intersects the x-y plane in the line l. Find the area of the finite triangular region enclosed by l, the x-axis and the y-axis.

Given two vectors p and q,

(i)     show that p\( \cdot \)p = \(|\)p\({|^2}\);

(ii)     hence, or otherwise, show that \(|\)p + q\({|^2}\) = \(|\)p\({|^2}\) + 2p\( \cdot \)q + \(|\)q\({|^2}\);

(iii)     deduce that \(|\)p + q\(|\) ≤ \(|\)p\(|\) + \(|\)q\(|\).

(a)     Show that a Cartesian equation of the line, \({l_1}\), containing points A(1, −1, 2) and B(3, 0, 3) has the form \(\frac{{x - 1}}{2} = \frac{{y + 1}}{1} = \frac{{z - 2}}{1}\).

(b)     An equation of a second line, \({l_2}\), has the form \(\frac{{x - 1}}{1} = \frac{{y - 2}}{2} = \frac{{z - 3}}{1}\). Show that the lines \({l_1}\) and \({l_2}\) intersect, and find the coordinates of their point of intersection.

(c)     Given that direction vectors of \({l_1}\) and \({l_2}\) are d\(_1\) and d\(_2\) respectively, determine d\(_1 \times \) d\(_2\).

(d)     Show that a Cartesian equation of the plane, \(\prod \), that contains \({l_1}\) and \({l_2}\) is \( - x - y + 3z = 6\).

(e)     Find a vector equation of the line \({l_3}\) which is perpendicular to the plane \(\prod \) and passes through the point T(3, 1, −4).

(f)     (i)     Find the point of intersection of the line \({l_3}\) and the plane \(\prod \).

         (ii)     Find the coordinates of \({{\text{T}}}'\), the reflection of the point T in the plane \(\prod \).

         (iii)     Hence find the magnitude of the vector \(\overrightarrow {{\text{TT}}'} \).

The acute angle between the vectors 3i − 4j − 5k and 5i − 4j + 3k is denoted by θ.

Find cos θ.

Let \(\alpha \) be the angle between the unit vectors a and b, where \(0 \leqslant \alpha \leqslant \pi \).

(a)     Express \(|\)a − b\(|\) and \(|\)a + b\(|\) in terms of \(\alpha \).

(b)     Hence determine the value of \(\cos \alpha \) for which \(|\)a + b\(|\) = 3 \(|\)a − b\(|\).

A triangle has vertices A(1, −1, 1), B(1, 1, 0) and C(−1, 1, −1) .

Show that the area of the triangle is \(\sqrt 6 \) .