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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

    $\displaystyle \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

    $\displaystyle {\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

    $\displaystyle \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

    $\displaystyle \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

    $\displaystyle \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

    $\displaystyle \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

    $\displaystyle {\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

    $\displaystyle {\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

    $\displaystyle {\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

    $\displaystyle {\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:

    $\displaystyle {\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:

    $\displaystyle {\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$ .

next up previous
Next: Cartesian Tensors Up: Vectors and Vector Fields Previous: Useful Vector Identities
Richard Fitzpatrick 2016-01-22