1. Let each of the vectors A = 5ax ay + 3az, B = 2ax + 2ay + 4az, and C = 3ay 4az extend outward from the origin of a Cartesian coordinate system to points A, B, and C, respectively. Find a unit vector directed from point A toward: (a) the origin; (b) point B; (c) a point equidistant from B and C on the line BC. (d) Find the length of the perimeter of the triangle ABC.

4. Given two points, M(2,5,3) and N(3,1,4): (a) find their separation; (b) find the distance from the origin to the midpoint of the line MN; (c) find a unit vector in the direction of RMN; (d) find the point of intersection of the line MN and the x = 0 plane.

10. Given points. P(2,5,1), Q(1,4,1), and T(5,0,2), find: (a) the vector RPQ; (b) a unit vector in the direction of RPQ; (c) the length of the perimeter of the triangle PQT; (d) the interior angle at Q; (e) the vector projection of RPQ on RPT; (f) the length of the altitude of the triangle that extends from Q perpendicularly to the opposite side or its extension.

11. Given points E(2,5,1), F(1,4,2), and G(3,2,4), find: (a) a unit vector directed from E towards F; (b) the angle between REF and REG; (c) the length of the perimeter of triangle EFG; (d) the scalar projection of REF on REG.

12. Express in Cartesian components: (a) the vector G extending from the origin to the midpoint of the line joining A(2,3,5) to B(6, 5,5); (b) the vector D extending from C(2,7,3) to the midpoint of the line joining A to B; (c) the component of RAB that is in the direction of RAC; (d) a unit vector in the direction of RBC.

14. Given the three points A(2,1,2), B(-1,1,4), and C(4,3,1), find: (a) the angle between RAB and RAC; (b) the (scalar) area of triangle ABC (c) a unit vector perpendicular to ABC.

17. Given the three points M(6,2,3), N(2,3,0), and P(4,6,5); find: (a) the area of the triangle they define; (b) a unit vector perpendicular to this triangular surface; (c) a unit vector bisecting the interior angle of the triangle at M.

19. Given the points P( = 5, = 60, z = 2) and Q( = 2, =110o, z= 1); (a) find the distance |RPQ|; (b) give a unit vector in Cartesian coordinates at P that is directed towards Q; (c) give a unit vector in cylindrical coordinates at P that is directed towards Q.

20. Find in cylindrical components: (a) a unit vector at P( = 5, = 53.13o, z = 2) in the direction of F = z cos a z sin a + az; (b) a unit vector at P parallel to ax; (c) a unit vector at Q( = 5, = 36.87o , z = 2) parallel to ax; (d) G = 2ax 4ay + 4az at P.

21. (a) Give the vector in Cartesian coordinates that extends from P( = 4, = 10o, z = 1) to Q( = 7, = 75, z = 4). (b) Give the vector in cylindrical coordinates at M(x = 5, y = 1, z = 2) that extends to N(2,4,6). (c) How far is it from A(110,60o,20) to B(30,125,10)?

23. Given points A(x = 2, y = 3, z = 1) and B( = 4, = 50o, z = 2), find a unit vector in cylindrical coordinates: (a) at point B directed toward point A; (b) at point A directed toward point B.

25. using the coordinate system names, give the vector at point A(2, 1, 3) that extends to B(1,3,4): (a) Cartesian; (b) cylindrical; (c) spherical.

28. Given the points M(r = 5, = 20o, = 120o) and N(r = 2, = 80o, = 30o): (a) find the distance from M to N; (b) give a unit vector in Cartesian coordinates at M that is directed toward N: (c) give a unit vector in spherical coordinates at M that is directed toward N.

30. (a) Give the vector in cartesian coordinates that extends from P(r = 4, = 20o, = 10o) to Q(r = 7, = 120o, = 75o). (b) Give the vector in spherical coordinates at M(x = 5, y = 1, z = 2) that extends to N(2,4,6). (c) How far is it from A(r = 110, = 30o, = 60o) to B(r = 30, = 75o, = 125o)?

2. A charge Qo = 1 nC is located in free space at P(a,0,0). Prepare a sketch of the magnitude of the force on Qo, as a function of a, 0 a 5m, produced by two other charges, Q1 = 1 C at (0,1,0) and Q2 = : (a) 1 C at (0,1,0); (b) 1 C at (0, 1,0).

4. A point charge, Q1 = 10 C, is located at P1(1,2,3) in free space, while Q2 = 5 C is at P2(1,2,10). (a) Find the vector force exerted on Q2 by Q1. (b) Find the coordinates of P3 at which a point charge Q3 experiences no force.

5. In free space, let Q1 = 10 nC be at P1(0,4,0), and Q2 = 20 nC be at O2(0,0,4). (a) Find E at the origin. (b) Where should a 30-nC point charge be located so that E=0 at the origin?

7. A point charge, QA = 1 C, is at A(0,0,1), and QB = 1 C is at B(0,0,1). Find E, E, and E at P(1,2,3).

11. Eight point charges of 1 nC each are located at the corners of a cube in free space that is 1 m on a side. Find |E| at the center of: (a) the cube; (b) a face; (c) an edge.

16. Let v = (x+2y+3z) C/m3 in the cubical region 0 x,y,z 1 mm, and v = 0 outside the cube. (a) What is the total charge contained within the cube? (b) Set up the volume integral that will give E(x,0,0) for x > 1 mm. Do not integrate.

17. Volume charge density is given as v = 10-5e-100rsin C/m3 for 0 r 1 cm, and v = 0 for r > 1 cm. Estimate E at r = 1 m, = 90o, = 0, by thinking in terms of a point charge.

18. A uniform volume charge density of 10 C/m3 is present in the spherical shell 0.9 < r < 1 m. and v = 0 elsewhere. (a) Find the Qtot, the total charge present. (b) In the next chapter we will see that this symmetrical charge distribution in free space produces an electric field for r > 1 m that is identical to the field that would be produced by a point charge Qtot at the origin. Find E in spherical coordinates for r > 1 m.