Let C be the curve which is the intersection of the half-cone S1 = {(x,y,z)|z=sqrt(x^2 + y^2} and the paraboloid S2 = {(x,y,z)|2z=3-x^2-y^2}. Find C.
a)Make a 3-D sketch to show S1, S2, and C
b) Show that at each point on C the normals to these two surfaces are perpendicular to each other.
To find the curve of intersection C between the half-cone S1 and the paraboloid S2, follow these steps:
a) 3-D Sketch:
To make a 3-D sketch to visualize S1, S2, and C, follow these steps:
1. Plot the half-cone S1: This half-cone has vertices at the origin (0, 0, 0) and extends upward along the positive z-axis. The half-cone is obtained by rotating the curve z = √(x^2 + y^2) around the z-axis.
2. Plot the paraboloid S2: This paraboloid has its vertex at the origin (0, 0, 0) and opens downward. The equation of the paraboloid is given by 2z = 3 - x^2 - y^2.
3. Find the points of intersection: Look for the points (x, y, z) that satisfy both the equation of the half-cone and the equation of the paraboloid (i.e., points that lie on both surfaces). These points will form the curve of intersection, which is C.
b) Normal Vectors:
To show that the normals to the two surfaces are perpendicular at each point on C, follow these steps:
1. Find the normal vector to S1: The normal vector to the surface S1 can be found by taking partial derivatives of the equation of S1 with respect to x, y, and z. The resulting vector will be perpendicular to the surface S1 at each point.
2. Find the normal vector to S2: Similarly, find the normal vector to the surface S2 by taking partial derivatives of the equation of S2 with respect to x, y, and z.
3. Evaluate the dot product: Once you have the normal vectors to S1 and S2, evaluate their dot product at each point of intersection on C. If the dot product is zero at each point, it confirms that the normal vectors are perpendicular to each other.
Using these steps, you can find C, plot the surfaces S1 and S2 in a 3-D sketch, and show that the normals to these surfaces are perpendicular at each point on C.