Question

The dot and cross products have many important uses in mathematics. Among the more useful is that they allow us to find projections of various sorts. For the dot product, this means that we can find out how muchof a vector v lies along a vector u. Similarly, the cross product gives us a way to construct a vector that is as much in another direction as possible given vectors v and u, the cross product gives a vector that is orthogonal to both. In this problem set we'll get some practice with coordinate systems, distances, and vectors, and explore some of the helpful geometric properties of the dot and cross products. Find the equations of the spheres with center (−1,3,4) that touch (i.e. meet at precisely one point) a) the yz-plane

(b) the plane x=2
(c) the sphere x² +4x+y² +z² +6z+9=0

Answer 1

The equations of the spheres with center (−1,3,4) that touch the yz-plane, plane x=2, and another sphere are derived using the distance from the center to these planes or by finding the sum of radii when touching another sphere.

Step-by-step explanation:

The spheres with center at (−1,3,4) that touch a given plane or sphere will have radii that are equal to the perpendicular distance from the center to the plane or the distance to the point of tangency on the other sphere. For part (a), since the yz-plane is defined by x=0, the distance from the center of the sphere to the yz-plane is simply the absolute value of the x-coordinate of the center, which is 1. So, the equation of the sphere is (x + 1)² + (y - 3)² + (z - 4)² = 1².

For part (b), the plane is given by x=2, and the distance from the center of the sphere to this plane is the absolute value of the center's x-coordinate minus 2, which is 3. Hence, the equation of the sphere is (x + 1)² + (y - 3)² + (z - 4)² = 3².

For part (c), we need to compare the given sphere's equation to the standard form (x - h)² + (y - k)² + (z - l)² = r², where (h, k, l) is the center and r is the radius. Completing the square for the given equation x² + 4x + y² + z² + 6z + 9 = 0 yields the center (−2,0,−3) and radius 2. To find the distance between the two centers, which will be equal to the sum of their radii (since they touch at a point), we use the distance formula, yielding a distance of 3, and hence the new sphere's radius. This gives us the sphere's equation (x + 1)² + (y - 3)² + (z - 4)² = 3², which is coincidentally the same as in part (b).