Let P(5, 1, −4) and Q(7, −3, 0) be two points in three-dimensional space.

Question

Let P(5, 1, −4)

 and Q(7, −3, 0)

 be two points in three-dimensional space.

(a) Find the distance between P and Q.

 

(b) Find an equation for the sphere whose center is P and for which the segment PQ

 is a radius of the sphere. (Let x, y,

 and z denote the standard coordinate variables.)

 

(c) The vector u has initial point P and terminal point Q. Express u both in component form and using the vectors i, j, and k.

Answer 1

(a) distance d beetween P and Q: PQ = √(7 - 5)² + (- 3 - 1)² + (0 - (- 4))² = √4 + 16 + 16 = √36 = 6

(b) Equation of Sphere: (x - 5)² + (y - 1)² + (z + 4)² = 36

(c) u = 2i - 4j + 4k