By Corral M.
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Extra resources for Vector Calculus
16. Write the normal form of the plane containing the lines from Exercise 10. For Exercises 17-18, find the distance d from the point Q to the plane P. 17. Q = (4, 1, 2), P : 3x − y − 5z + 8 = 0 18. Q = (0, 2, 0), P : −5x + 2y − 7z + 1 = 0 For Exercises 19-20, find the line of intersection (if any) of the given planes. 19. x + 3y + 2z − 6 = 0, 2x − y + z + 2 = 0 20. 3x + y − 5z = 0, x + 2y + z + 4 = 0 B x−6 = y + 3 = z with the plane 4 x + 3y + 2z − 6 = 0. ) 21. Find the point(s) of intersection (if any) of the line 40 CHAPTER 1.
19. Let Q = (x0 , y0 , z0 ) be a point in 3 , and let P be a plane with normal form ax + by + cz + d = 0 that does not contain Q. 27) Proof: Let R = (x, y, z) be any point in the plane P (so that ax + by + cz + d = 0) and −−→ let r = RQ = (x0 − x, y0 − y, z0 − z). Then r 0 since Q does not lie in P. From the normal form equation for P, we know that n = (a, b, c) is a normal vector for P. Now, any plane divides 3 into two disjoint parts. Assume that n points toward the side of P where the point Q is located.
Find the cylindrical coordinates of that point of intersection. 11. Let P1 and P2 be points whose spherical coordinates are (ρ1 , θ1 , φ1 ) and (ρ2 , θ2 , φ2 ), respectively. Let v1 be the vector from the origin to P1 , and let v2 be the vector from the origin to P2 . For the angle γ between v1 and v2 , show that cos γ = cos φ1 cos φ2 + sin φ1 sin φ2 cos( θ2 − θ1 ). This formula is used in electrodynamics to prove the addition theorem for spherical harmonics, which provides a general expression for the electrostatic potential at a point due to a unit charge.