- The position vectors of the four points , , , and are , , , and , respectively. Express , , , and in terms of and .
- Prove the trigonometric law of sines

using vector methods. Here, , , and are the three angles of a plane triangle, and , , and the lengths of the corresponding opposite sides. - Demonstrate using vectors that the diagonals of a parallelogram bisect one another. In addition, show that if the diagonals of a quadrilateral bisect one another then it is a parallelogram.
- From the inequality

deduce the triangle inequality

- Find the scalar product
and the vector product
when
- , ,
- , .

- Which of the following statements regarding the three general vectors , , and are true?
- .
- .
- .
- implies that .
- implies that .
- .

- Prove that the length of the shortest straight-line from point to the
straight-line joining points and is

- Identify the following surfaces:
- ,
- ,
- ,
- .

- Let , , and be coplanar vectors related via

where , , and are not all zero. Show that the condition for the points with position vectors , , and to be colinear is

- If , , and are any vectors, demonstrate that
,
,
and
are coplanar provided that
, where , , and are scalars.
Show that this condition is satisfied when is perpendicular to , to , and to .
- The vectors , , and are non-coplanar, and
form a non-orthogonal vector base. The vectors , ,
and , defined by

plus cyclic permutations, are said to be*reciprocal vectors*. Show that

plus cyclic permutations. - In the notation of the previous question, demonstrate that the plane passing
through points
, , and
is normal to the direction of the vector

In addition, show that the perpendicular distance of the plane from the origin is . - Evaluate
for

around the square whose sides are , , , . - Find the gradients of the following scalar functions of the position vector
:
- ,
- ,