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 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 straightline from point
to the
straightline joining points
and
is
 Identify the following surfaces:

,

,

,

.
Here,
is the position vector,
,
,
, and
are positive
constants, and
is a fixed unit vector.
 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 noncoplanar, and
form a nonorthogonal 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 exercise, 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
,
,
,
.
 Consider the following vector field:
Is this field conservative? Is it solenoidal? Is it irrotational? Justify your answers. Calculate
, where the curve
is a unit circle in the

plane, centered on the origin, and the
direction of integration is clockwise looking down the
axis.
 Consider the following vector field:
Is this field conservative? Is it solenoidal? Is it irrotational? Justify your answers. Calculate the flux of
out of a unit sphere centered
on the origin.
 Find the gradients of the following scalar functions of the position vector
:

,


,

Here,
is a fixed vector.
 Find the divergences and curls of the following vector fields:

,

,

,

.
Here,
and
are fixed vectors.
 Calculate
when
. Find
if
.
Next: Cartesian Tensors
Up: Vectors and Vector Fields
Previous: Useful Vector Identities
Richard Fitzpatrick
20160122