   Chapter 12.5, Problem 14E

Chapter
Section
Textbook Problem

# Is the line through (−2, 4, 0) and (1, 1, 1) perpendicular to the line through (2, 3, 4) and (3, −1, −8)?

To determine

To find: Whether the line through the points (2,4,0) and (1,1,1) is perpendicular to the line through the points (2,3,4) and (3,1,8) or not.

Explanation

The condition for two lines to be perpendicular to each other, the dot product of the direction vectors of the two lines is zero.

Consider the direction vector of line through the points (2,4,0) and (1,1,1) is v1 and the direction vector of line through the points (2,3,4) and (3,1,8) is v2 .

The lines to be perpendicular to each other, v1v2=0 .

Formula:

Write the expression to find direction vector of the line through the points (x1,y1,z1) and (x2,y2,z2) .

v=(x2x1),(y2y1),(z2z1) (1)

Calculation of direction vector of the line through the points (2,4,0) and (1,1,1) :

In equation (1), 2 for x1 , 4 for y1 , 0 for z1 , 1 for x2 , 1 for y2 , and 1 for z2 .

v1=[1(2)],(14),(10)=3,3,1

The direction vector v1 is also written as follows.

v1=3i3j+k

Calculation of direction vector of the line through the points (2,3,4) and (3,1,8) :

In equation (1), 2 for x1 , 3 for y1 , 4 for z1 , 3 for x2 , 1 for y2 , and 8 for z2

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