Multivar Notes 110926
Consider the surfaces
x^2 + y^2 + z^2 = 25 (a sphere of radius 5 centered at the origin)
4 x^2 + 9 y^2 - z^2 = 36 (a hyperboloid of two sheets, one symmetric about the y axis and the other about the x axis).
We investigate their intersection, beginning with their intersection in the planes x = 0, x = 2 and x = 4.
In the plane x = 0 the sphere forms a circle of radius 5. The hyperboloid has vertices (y, z) = (+-2, 0) and asymptotes z +- 3 y. It intersects the circle once in each quadrant.
In the plane x = 2 the sphere forms a circle of radius approximately 4.6. The hyperboloid has vertices (y, z) = (+-1.5, 0), very approximately, and asymptotes z +- 3 y. It intersects the circle once in each quadrant, and the points of intersection are closer to the center of the circle than in the x = 0 plane.
In the plane x = 4 the sphere forms a circle of radius 3. The hyperboloid has vertices (y, z) = (0, +-4.9), very approximately, and does not intersect the sphere.
The curve of intersection for the two surfaces lies between the planes x = -3 and x = 3. The equation of the hyperbola in the plane x = c is
y^2 / a^2 - x^2 / b^2 = 1
with a = 1/3 sqrt( 36 - 4 c^2) and b = sqrt(36 - 4 c^2)
and the equation of the circle in this plane is
x^2 + y^2 = 36 - c^2.
If we want to further visualize the intersection, consider the following:
At any intersection point the coordinates of the three points must satisfy both equations.
If we add the two equations
x^2 + y^2 + z^2 = 25
and
4 x^2 + 9 y^2 - z^2 = 36
we will eliminate z, obtaining
5 x^2 + 10 y^2 = 51.
In the x-y plane this is an ellipse, with major and minor axes of about 3.2 (in the x direction) and 2.3 (in the y direction). The intersection of the hyperboloid with the sphere therefore lies above and below this ellipse, forming a curve in xyz space. This is much easier to visualize than the curve of intersection of the hyperboloid.
Summary of main ideas of Chapter 10:
A vector function F(t) = x(t) i + y(t) j + z(t) k defines a curve in xyz space.
F (t) has a derivative F ' (t) = x ' (t) i + y ' (t) j + z ' (t) k. If x, y and z are considered to be functions of clock time t, this derivative is the velocity function associated with the motion. In any case this derivative is tangent to the curve.
F '' (t) has components parallel and perpendicular to those of F ' (t), easily found by projecting the former onto the latter, then subtracting the vector projection from the former. Details of this process are not difficult to understand, provided you understand the projection. If t is interpreted as clock time, then the parallel component of F '' is the acceleration in the direction of motion (which causes the particle to speed up or slow down) and the perpendicular component is the acceleration perpendicular to the direction of motion (identified with something very much like a centripetal acceleration, i.e., an acceleration which changes direction but not speed).
Practice Ch. 9 Test
1. Let u = -2 i + 4 j, v = -3 i - 5 j, w = i - 4 j. Find a vector of length 6 with the same direction as u - 2 v + 3 w.
2. Find the equation of the sphere with center ( -3, -1, 4) and tangent the the plane 2 x + y - 3 z = 4.
3. Find the work performed when the force F = 2 c i - 3 c j + 7 c k is applied to an object moving along the line from P(2,1,3) to Q(-3,-6,-9). Find the value of c if the work done is 128.
4. Find the scalar and vector projections of u = 4 i - 2 j + 3 k on v = - 2 j + 5 k, as well the components of u in in the direction of v and perpendicular to this direction.
5. Tell whether the line represented by the parametric equations x = 3 - t, y = -4 t , z = 2 + 3 t and the line represented by x = 7- 4t, y = -1 + 3 t, z = -5 +2 t coincide, intersect, are parallel, or if they are skew. If they intersect, give the point of intersection.
6. Find the distance between the point P(3,-2, 5) and the plane x + y - 2 z = 6.
7. Describe and sketch the quadric surface 4 z^2 / 9 + 2 x^2 + y^2 / 16 = 25. What is the equation of intersection of this surface with the plane x = c, and how does the graph of the intersection change with increasing values of c?