Exercises

1. The position vectors of the four points $A$ , $B$ , $C$ , and $D$ are ${\bf a}$ , ${\bf b}$ , $3\,{\bf a}+2\,{\bf b}$ , and ${\bf a}-3\,{\bf b}$ , respectively. Express $\stackrel{\displaystyle \rightarrow}{AC}$ , $\stackrel{\displaystyle \rightarrow}{DB}$ , $\stackrel{\displaystyle \rightarrow}{BC}$ , and $\stackrel{\displaystyle \rightarrow}{CD}$ in terms of ${\bf a}$ and ${\bf b}$ .
2. Prove the trigonometric law of sines
   $$\frac{\sin a}{A} = \frac{\sin b}{B} = \frac{\sin c}{C}$$
   using vector methods. Here, $a$ , $b$ , and $c$ are the three angles of a plane triangle, and $A$ , $B$ , and $C$ the lengths of the corresponding opposite sides.

3. 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.

4. From the inequality
   $${\bf a}\cdot{\bf b}= \vert{\bf a}\vert\,\vert{\bf b}\vert\,\cos\theta\leq \vert{\bf a}\vert\,\vert{\bf b}\vert$$
   deduce the triangle inequality
   $$\vert{\bf a} + {\bf b}\vert\leq \vert{\bf a}\vert+\vert{\bf b}\vert.$$

5. Find the scalar product ${\bf a} \cdot {\bf b}$ and the vector product ${\bf a}\times{\bf b}$ when
   1. ${\bf a} = {\bf e}_x + 3\,{\bf e}_y-{\bf e}_z$ , ${\bf b} = 3\,{\bf e}_x+ 2\,{\bf e}_y+{\bf e}_z$ ,
   2. ${\bf a} = {\bf e}_x - 2\,{\bf e}_y+{\bf e}_z$ , ${\bf b} = 2\,{\bf e}_x+ {\bf e}_y+{\bf e}_z$ .

6. Which of the following statements regarding the three general vectors ${\bf a}$ , ${\bf b}$ , and ${\bf c}$ are true?
   1. ${\bf c}\cdot ({\bf a}\times {\bf b}) = ({\bf b}\times {\bf a})\cdot{\bf c}$ .
   2. ${\bf a}\times ({\bf b}\times {\bf c})= ({\bf a}\times {\bf b})\times {\bf c}$ .
   3. ${\bf a}\times ({\bf b}\times {\bf c}) = ({\bf a}\cdot{\bf c})\,{\bf b} - ({\bf a}\cdot{\bf b})\,{\bf c}$ .
   4. ${\bf d} = \lambda\,{\bf a} + \mu\,{\bf b}$ implies that $({\bf a}\times {\bf b})\cdot {\bf d} = 0$ .
   5. ${\bf a}\times {\bf c} = {\bf b}\times {\bf c}$ implies that ${\bf c}\cdot{\bf a} - {\bf c}\cdot{\bf b} = c\,\vert{\bf a}-{\bf b}\vert$ .
   6. $({\bf a}\times {\bf b})\times ({\bf c}\times {\bf b}) = [{\bf b}\cdot({\bf c}\times {\bf a})]\,{\bf b}$ .

7. Prove that the length of the shortest straight-line from point ${\bf a}$ to the straight-line joining points ${\bf b}$ and ${\bf c}$ is
   $$\frac{\vert{\bf a}\times {\bf b} + {\bf b}\times {\bf c} + {\bf c}\times {\bf a}\vert}{\vert{\bf b}-{\bf c}\vert}.$$

8. Identify the following surfaces:
   1. $\vert{\bf r}\vert = a$ ,
   2. ${\bf r}\cdot{\bf n} = b$ ,
   3. ${\bf r}\cdot{\bf n} = c\,\vert{\bf r}\vert$ ,
   4. $\vert{\bf r} -({\bf r}\cdot{\bf n})\,{\bf n}\vert = d$ .
   Here, ${\bf r}$ is the position vector, $a$ , $b$ , $c$ , and $d$ are positive constants, and ${\bf n}$ is a fixed unit vector.

9. Let ${\bf a}$ , ${\bf b}$ , and ${\bf c}$ be coplanar vectors related via
   $$\alpha\,{\bf a} + \beta\,{\bf b} + \gamma\,{\bf c} = {\bf0},$$
   where $\alpha$ , $\beta$ , and $\gamma$ are not all zero. Show that the condition for the points with position vectors $u\,{\bf a}$ , $v\,{\bf b}$ , and $w\,{\bf c}$ to be colinear is
   $$\frac{\alpha}{u} +\frac{\beta}{v} + \frac{\gamma}{w} = 0.$$

10. If ${\bf p}$ , ${\bf q}$ , and ${\bf r}$ are any vectors, demonstrate that ${\bf a}={\bf q} + \lambda\,{\bf r}$ , ${\bf b} = {\bf r}+\mu\,{\bf p}$ , and ${\bf c} = {\bf p} + \nu\,{\bf q}$ are coplanar provided that $\lambda\,\mu\,\nu=-1$ , where $\lambda$ , $\mu$ , and $\nu$ are scalars. Show that this condition is satisfied when ${\bf a}$ is perpendicular to ${\bf p}$ , ${\bf b}$ to ${\bf q}$ , and ${\bf c}$ to ${\bf r}$ .

11. The vectors ${\bf a}$ , ${\bf b}$ , and ${\bf c}$ are non-coplanar, and form a non-orthogonal vector base. The vectors ${\bf A}$ , ${\bf B}$ , and ${\bf C}$ , defined by
    $${\bf A} = \frac{{\bf b}\times {\bf c}}{{\bf a}\cdot{\bf b}\times {\bf c}},$$
    plus cyclic permutations, are said to be reciprocal vectors. Show that
    $${\bf a} = ({\bf B}\times {\bf C})/({\bf A}\cdot{\bf B}\times {\bf C}),$$
    plus cyclic permutations.

12. In the notation of the previous exercise, demonstrate that the plane passing through points ${\bf a}/\alpha$ , ${\bf b}/\beta$ , and ${\bf c}/\gamma$ is normal to the direction of the vector
    $${\bf h} = \alpha\,{\bf A} + \beta\,{\bf B} + \gamma\,{\bf C}.$$
    In addition, show that the perpendicular distance of the plane from the origin is $\vert{\bf h}\vert^{-1}$ .

13. Evaluate $\oint {\bf A}\cdot d{\bf r}$ for
    $${\bf A} = \frac{x\,{\bf e}_x + y\,{\bf e}_y}{\sqrt{x^{\,2} + y^{\,2}}}$$
    around the square whose sides are $x=0$ , $x=a$ , $y=0$ , $y=a$ .

14. Consider the following vector field:
    $${\bf A}({\bf r}) = (8\,x^{\,3}+3\,x^{\,2}\,y^{\,2},\,2\,x^{\,3}\,y+6\,y,\,6).$$
    Is this field conservative? Is it solenoidal? Is it irrotational? Justify your answers. Calculate $\oint_C {\bf A}\cdot d{\bf r}$ , where the curve $C$ is a unit circle in the $x$ -$y$ plane, centered on the origin, and the direction of integration is clockwise looking down the $z$ -axis.

15. Consider the following vector field:
    $${\bf A}({\bf r}) = (3\,x\,y^{\,2}\,z^{\,2}-y^{\,2},\,-y^{\,3}\,z^{\,2}+x^{\,2}\,y,\,3\,x^{\,2}-x^{\,2}\,z).$$
    Is this field conservative? Is it solenoidal? Is it irrotational? Justify your answers. Calculate the flux of ${\bf A}$ out of a unit sphere centered on the origin.

16. Find the gradients of the following scalar functions of the position vector ${\bf r}=(x,\,y,\,z)$ :
    1. ${\bf k}\cdot{\bf r}$ ,
    2. $\vert{\bf r}\vert^{\,n},$
    3. $\vert{\bf r}-{\bf k}\vert^{-n}$ ,
    4. $\cos({\bf k}\cdot {\bf r}).$
    Here, ${\bf k}$ is a fixed vector.

17. Find the divergences and curls of the following vector fields:
    1. ${\bf k}\times {\bf r}$ ,
    2. $\vert{\bf r}\vert^{\,n}\,{\bf r}$ ,
    3. $\vert{\bf r} - {\bf k}\vert^{n}\,({\bf r}-{\bf k})$ ,
    4. ${\bf a}\,\cos({\bf k}\cdot {\bf r})$ .
    Here, ${\bf k}$ and ${\bf a}$ are fixed vectors.

18. Calculate $\nabla^{\,2}\phi$ when $\phi=f(\vert{\bf r} \vert)$ . Find $f$ if $\nabla^{\,2}\phi=0$ .