   Chapter 12.3, Problem 25E

Chapter
Section
Textbook Problem

# Use vectors to decide whether the triangle with vertices P(l, −3, −2), Q(2, 0, −4), and R{6, −2, −5) is right-angled.

To determine

Whether the triangle with vertices is right-angled.

Explanation

Given:

P(1,3,2),Q(2,0,4) and R(6,2,5) .

Formula:

Consider a general expression to find the dot product between two three-dimensional vectors.

PQPR=a1,a2,a3b1,b2,b3 (1)

PQPR=a1b1+a2b2+a3b3

Condition for orthogonal:

Two vectors a and b are orthogonal if and only if ab=0 .

Find PQ .

PQ=Q(2,0,4)P(1,3,2)=21,0(3),(4)(2)=1,3,2

Find PR .

PR=R(6,2,5)P(1,3,2)=61,(2)(3),(5)(2)=5,1,3

In equation (1), substitute 1,3,2 for PQ and 5,1,3 for PR .

PQPR=1,3,25,1,3=(1)(5)+(3)(1)+(2)(3)=5+3+6=14

The dot product between PQ and PR vectors doesn’t satisfies the condition of orthogonal

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