Exercises

1. 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 a plane triangle's three angles, and $A$, $B$, and $C$ the lengths of the corresponding opposite sides.

2. 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.

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

4. 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.

5. 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 lie on the same straight-line is
   
   $$\frac{\alpha}{u} +\frac{\beta}{v} + \frac{\gamma}{w} = 0.$$
   
   

6. 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 ${\bf r}$.

7. The vectors ${\bf a}$, ${\bf b}$, and ${\bf c}$ are not 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.

8. In the notation of the previous question, 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 $\vert{\bf h}\vert^{-1}$.
9. Find the gradients of the following 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.