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# Exercises

1. The position vectors of the four points , , , and are , , , and , respectively. Express , , , and in terms of and .
2. Prove the trigonometric law of sines

using vector methods. Here, , , and are the three angles of a plane triangle, and , , and 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

deduce the triangle inequality

5. Find the scalar product and the vector product when
1. , ,
2. , .

6. Which of the following statements regarding the three general vectors , , and are true?
1. .
2. .
3. .
4. implies that .
5. implies that .
6. .

7. Prove that the length of the shortest straight-line from point to the straight-line joining points and is

8. Identify the following surfaces:
1. ,
2. ,
3. ,
4. .
Here, is the position vector, , , , and are positive constants, and is a fixed unit vector.

9. Let , , and be coplanar vectors related via

where , , and are not all zero. Show that the condition for the points with position vectors , , and to be colinear is

10. If , , and are any vectors, demonstrate that , , and are coplanar provided that , where , , and are scalars. Show that this condition is satisfied when is perpendicular to , to , and to .

11. The vectors , , and are non-coplanar, and form a non-orthogonal vector base. The vectors , , and , defined by

plus cyclic permutations, are said to be reciprocal vectors. Show that

plus cyclic permutations.

12. In the notation of the previous exercise, demonstrate that the plane passing through points , , and is normal to the direction of the vector

In addition, show that the perpendicular distance of the plane from the origin is .

13. Evaluate for

around the square whose sides are , , , .

14. Consider the following vector field:

Is this field conservative? Is it solenoidal? Is it irrotational? Justify your answers. Calculate , where the curve is a unit circle in the - plane, centered on the origin, and the direction of integration is clockwise looking down the -axis.

15. Consider the following vector field:

Is this field conservative? Is it solenoidal? Is it irrotational? Justify your answers. Calculate the flux of out of a unit sphere centered on the origin.

16. Find the gradients of the following scalar functions of the position vector :
1. ,
2. ,
Here, is a fixed vector.

17. Find the divergences and curls of the following vector fields:
1. ,
2. ,
3. ,
4. .
Here, and are fixed vectors.

18. Calculate when . Find if .

Next: Cartesian Tensors Up: Vectors and Vector Fields Previous: Useful Vector Identities
Richard Fitzpatrick 2016-03-31