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\documentclass[10pt,oneside,a4paper]{article} \usepackage[english]{babel} \usepackage{textcomp} \usepackage{tikz} \usetikzlibrary{arrows} \addtolength{\oddsidemargin}{-.875in} \addtolength{\evensidemargin}{-.875in} \addtolength{\textwidth}{1.75in} \addtolength{\topmargin}{-.875in} \addtolength{\textheight}{1.875in} \begin{document} \begin{center} PROBLEMS OF VECTOR ANALYSIS\\ {\bf BY MURRAY R SPIEGEL}\\ \end{center} 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:\\ \mathbf{A}, 10m northwest; \mathbf{B}, 20m 30 \textdegree{} north of east; \mathbf{C}, 35m due south. \item Show that addition of vectors is commutative, i.e \mathbf{A}+\mathbf{B}=\mathbf{B}+\mathbf{A}. \item Show that addition of vectors is associative, i.e. \mathbf{A}+(\mathbf{B}+\mathbf{C})=(\mathbf{A}+\mathbf{B})+\mathbf{C}. \item Forces \mathbf{F}_{1}, \mathbf{F}_{2}, ... , \mathbf{F}_{6} act as shown on an object P. What force is needed to prevent P from moving? \begin{center} \begin{tikzpicture}[scale=0.4] \draw [arrows={latex'-latex'}] (2.5,5) – (5,5) – (5,2.5); \draw [arrows={latex'-latex'}] (0,10) – (5,5) – (11,1); \draw [arrows={latex'-latex'}] (8,7) – (5,5) – (9,10); \node at (7.0,5.6) {\small \mathbf{F}_{1}}; \node at (8.0,2.5) {\small \mathbf{F}_{2}}; \node at (5.55,3.7) {\small \mathbf{F}_{3}}; \node at (3.8,4.5) {\small \mathbf{F}_{4}}; \node at (2.0,7.0) {\small \mathbf{F}_{5}}; \node at (6.5,8.0) {\small \mathbf{F}_{6}}; \node at (5.0,5.8) {\small P}; \end{tikzpicture} \end{center} \item Given vectors \mathbf{A}, \mathbf{B} and \mathbf{C}, construct (a) \mathbf{A} - \mathbf{B} + 2 \mathbf{C} (b) 3\mathbf{C} - \frac{1}{2} ( 2 \mathbf{A} - \mathbf{B}). \begin{center} \begin{tikzpicture}[scale=0.25] \draw [arrows={-latex'}] (0,2) – (6,8); \draw [arrows={-latex'}] (3.8,3.1) – (11,0); \draw [arrows={-latex'}] (12,8.5) – (14.0,2.5); \node at (2.5,5.5) {\small \mathbf{A}}; \node at (13.6,6.4) {\small \mathbf{B}}; \node at (6.9,1.0) {\small \mathbf{C}}; \end{tikzpicture} \end{center} \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 \mathbf(a) and \mathbf{b}, find an expression for any vector \mathbf{r} lying in the plane determined by \mathbf(a) and \mathbf{b}. \item Given three non-coplanar vectors \mathbf(a), \mathbf(b) and \mathbf{c}, find an expression for any vector \mathbf{r} in three dimensional space. \item Prove that if \mathbf{a} and \mathbf{b} are non-collinear then x\mathbf{a}+y\mathbf{b}=\mathbf{0} implies x=y=0, \item Prove that if x_1\mathbf{a} + y_1\mathbf{b}=x_2\mathbf{a}+y_2\mathbf{b}, where \mathbf{a} and \mathbf{b} are non-collinear, then x_1=x_2 and y_1=y_2. \item Prove that if \mathbf{a}, \mathbf{b} and \mathbf{c} are non-coplanar then x\mathbf{a}+y\mathbf{b}+z\mathbf{c}=0 implies x=y=z=0. \item Prove that if x_1\mathbf{a} + y_1\mathbf{b}+z_1\mathbf{c}=x_2\mathbf{a}+y_2\mathbf{b}=z_2\mathbf{c}, where \mathbf{a}, \mathbf{b} and \mathbf{c} are non-collinear, then x_1=x_2, y_1=y_2 and z_1=z_2. \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 P_1, P_2 and P_3 be points fixed relative to an origin O and let \mathbf{r}_1, \mathbf{r}_2 and \mathbf{r}_3 be position vectors from O to each point. Show that if the vector equation a_1\mathbf{r}_1+a_2\mathbf{r}_2+a_3\mathbf{r}_3=\mathbf{0} holds with respect to origin O then it will hold with respect to any other origin O' if and only if a_1+a_2+a_3=0. \item Find the equation of a straight line which passes through two given points A and B having position vectors \mathbf{a} and \mathbf{b} with respect to an origin O. \item (a) Find the position vectors \mathbf{r}_1 and \mathbf{r}_2 for the points P(2,4,3) and Q(1,-5,2) of a rectangular co-ordinate system in terms of the unit vectors \mathbf{i}, \mathbf{j}, \mathbf{k}. (b) Determine graphically and analytically the resultant of these position vectors. \item Prove that the magnitude A of the vector \mathbf{A}=A_1\mathbf{i}+A_2\mathbf{j}+A_3\mathbf{k} is A=\sqrt{A_1^2+A_2^2+A_3^2}. \item Given \mathbf{r}_1=3\mathbf{i}-2\mathbf{j}+\mathbf{k}, \mathbf{r}_2=2\mathbf{i}-4\mathbf{j}-3\mathbf{k}, \mathbf{r}_3=-\mathbf{i}+2\mathbf{j}+2\mathbf{k}, find the magnitudes of (a) \mathbf{r}_2, (b) \mathbf{r}_1+\mathbf{r}_2+\mathbf{r}_3, (c) 2\mathbf{r}_1-3\mathbf{r}_2-5\mathbf{r}_3. \item If \mathbf{r}_1=2\mathbf{i}-\mathbf{j}+\mathbf{k}, \mathbf{r}_2=\mathbf{i}+3\mathbf{j}-2\mathbf{k}, \mathbf{r}_3=-2\mathbf{i}+\mathbf{j}-3\mathbf{k} and \mathbf{r}_4=3\mathbf{i}+2\mathbf{j}+5\mathbf{k}, find scalars a,b,c such that \mathbf{r}_4=a\mathbf{r}_1+b\mathbf{r}_2+c\mathbf{r}_3. \item Find a unit vector parallel to the resultant of vectors \mathbf{r}_1=2\mathbf{i}+4\mathbf{j}-5\mathbf{k}, \mathbf{r}_2=\mathbf{i}+2\mathbf{j}+3\mathbf{k}. \item Determine the vector having initial point P(x_1,y_1,z_1) and terminal point Q(x_2,y_2,z_2) and find its magnitude. \item Forces \mathbf{A}, \mathbf{B} and \mathbf{C} acting on an object are given in terms of their components by the vector equations \mathbf{A}=A_1\mathbf{i}+A_2\mathbf{j}+A_3\mathbf{k}, \mathbf{B}=B_1\mathbf{i}+B_2\mathbf{j}+B_3\mathbf{k}, \mathbf{C}=C_1\mathbf{i}+C_2\mathbf{j}+C_3\mathbf{k}. Find the magnitude of the resultant of these forces. \item Determine the angles \alpha, \beta and \gamma which the vector \mathbf{r}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k} makes with the positive directions of the coordinate axes and show that \begin{center} cos^2(\alpha)+cos^2(\beta)+cos^2(\gamma)=1. \\ \end{center} (The numbers cos(\alpha), cos(\beta), cos(\gamma) are called the \emph{direction cosines} of the vector \mathbf{r}.) \item Determine a set of equations for the straight line passing through the points P(x_1,y_1,z_1) and Q(x_2,y_2,z_2). \item Given the scalar field defined by \phi(x,y,z)=3x^2z-xy^3+5 find \phi at the points \\ (a) (0,0,0), (b) (1,-2,2), (c) (-1,-2,-3). \item Graph the vector fields defined by: \\ (a) \mathbf{V}(x,y)=x\mathbf{i}+y\mathbf{j}, (b) \mathbf{V}(x,y)=-x\mathbf{i}-y\mathbf{j}, (c) \mathbf{V}(x,y,z)=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}. \end{enumerate} \end{document}


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