Time for some \TeX:
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PROBLEMS OF VECTOR ANALYSIS\\
{\bf BY MURRAY R SPIEGEL}\\
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Chapter 1 — {\bf VECTORS and SCALARS}
\begin{enumerate}
\item State which of the following are scalars and which are vectors.
\begin{tabbing}
(a) weight \= (c) specific heat
\= (e) density
\= (g) volume
\= (i) speed \\
(b) calorie \> (d) momentum \> (f) energy \> (h) distance \> (j) magnetic field intensity \\
\end{tabbing}
\item \begin{tabbing} Represent graphically \= (a) a force of 10N in a direction 30 \textdegree{} north of east \\
\>(b) a force of 15N in a direction 30 \textdegree{} east of north. \\ \end{tabbing}
\item An automobile travels 3km due north, then 5km northeast. Represent these displacements graphically and determine the resultant displacement (a) graphically, (b) analytically.
\item Find the sum or resultant of the following displacements:\\
, 10m northwest;
, 20m 30 \textdegree{} north of east;
, 35m due south.
\item Show that addition of vectors is commutative, i.e
.
\item Show that addition of vectors is associative, i.e.
.
\item Forces
act as shown on an object
. What force is needed to prevent
from moving?
\begin{center}
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\item Given vectors
,
and
, construct (a)
(b)
.
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\item An aeroplane moves in a northwesterly direction at 125km/h relative to the ground, owing to the fact that there is a westerly wind of 50km/h relative to the ground. How quickly and in what direction would the plane have travelled if there were no wind?
\item Given two non-collinear vectors
and
, find an expression for any vector
lying in the plane determined by
and
.
\item Given three non-coplanar vectors
,
and
, find an expression for any vector
in three dimensional space.
\item Prove that if
and
are non-collinear then
implies
\item Prove that if
, where
and
are non-collinear, then
and
.
\item Prove that if
,
and
are non-coplanar then
implies
.
\item Prove that if
, where
,
and
are non-collinear, then
,
and
.
\item Prove that the diagonals of a parallelogram bisect one another.
\item If the midpoints of the consecutive sides of any quadrilateral are connected by straight lines, prove that the resulting quadrilateral is a parallelogram.
\item Let
,
and
be points fixed relative to an origin
and let
,
and
be position vectors from
to each point. Show that if the vector equation
holds with respect to origin
then it will hold with respect to any other origin
if and only if
.
\item Find the equation of a straight line which passes through two given points
and
having position vectors
and
with respect to an origin
.
\item (a) Find the position vectors
and
for the points
and
of a rectangular co-ordinate system in terms of the unit vectors
. (b) Determine graphically and analytically the resultant of these position vectors.
\item Prove that the magnitude
of the vector
is
.
\item Given
,
,
, find the magnitudes of (a)
, (b)
, (c)
.
\item If
,
,
and
, find scalars
such that
.
\item Find a unit vector parallel to the resultant of vectors
.
\item Determine the vector having initial point
and terminal point
and find its magnitude.
\item Forces
,
and
acting on an object are given in terms of their components by the vector equations
. Find the magnitude of the resultant of these forces.
\item Determine the angles
,
and
which the vector
makes with the positive directions of the coordinate axes and show that \begin{center}
. \\ \end{center}
(The numbers
are called the \emph{direction cosines} of the vector
.)
\item Determine a set of equations for the straight line passing through the points
and
.
\item Given the scalar field defined by
find
at the points \\ (a)
, (b)
, (c)
.
\item Graph the vector fields defined by: \\ (a)
, (b)
, (c)
.
\end{enumerate}
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