   Chapter 12.1, Problem 11E

Chapter
Section
Textbook Problem

# Determine whether the points lie on a straight line.(a) A(2, 4, 2), B(3, 7, −2), C(1, 3, 3)(b) D(0,−5, 5), E(1, −2, 4), F(3, 4, 2)

(a)

To determine

Whether the points lie on a straight line or not.

Explanation

The points are A(2,4,2) , B(3,7,2) , and C(1,3,3) .

Consider the distance between the points A and B to be |AB| , the distance between the points B and C is |BC| , and the distance between the points A and C is |AC| .

For the three points to be on a straight line, the sum of the two shortest distances must be equal to the longest distance.

Formula:

Write the expression to find the distance between two points P1(x1,y1,z1) and P2(x2,y2,z2) .

|P1P2|=(x2x1)2+(y2y1)2+(z2z1)2 (1)

Calculation of distance between the points A and B:

In equation (1), substitute 2 for x1 , 4 for y1 , 2 for z1 , 3 for x2 , 7 for y2 , and 2 for z2 to find the distance between the points A and B.

|AB|=(32)2+(74)2+(22)2=1+9+16=26=5.0990

The distance between the points A and B is 5.0990.

Calculation of the distance between the points B and C:

In equation (1), substitute 3 for x1 , 7 for y1 , 2 for z1 , 1 for x2 , 3 for y2 , and 3 for z2 to find the distance between the points B and C

(b)

To determine

Whether the points lie on a straight line or not.

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