   Chapter 12.2, Problem 47E

Chapter
Section
Textbook Problem

# If r = ⟨x, y, z⟩ and r0 = ⟨x0, y0, z0⟩, describe the set of all points (x, y, z) such that | r − r0 | = 1.

To determine

To describe: All set of points for condition |rr0|=1.

Explanation

Given:

Two three-dimensional vectors r=x,y,z and r0=x0,y0,z0.

Definitions:

Subtraction of vectors:

Consider the two three-dimensional vectors such as a=a1,a2,a3 and b=b1,b2,b3.

The vector subtraction of two vectors (ab) is,

(ab)=a1,a2,a3b1,b2,b3=a1b1,a2b2,a3b3

Formula used:

Consider the expression for magnitude of vector a=a1,a2,a3(|a|).

|a|=a12+a22+a32 (1)

Write the expression for sphere with center C(h,k,l) and radius r.

(xh)2+(yk)2+(zl)2=r2 (2)

Write the expression for relation between vectors r and r0.

|rr0|=1

From definition, substitute x,y,z for r and x0,y0,z0 for r0,

|x,y,zx0,y0,z0|=1

|xx0,yy0,zz0|=1 (3)

Find the value of |xx0,yy0,zz0| by using equation (1)

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