   Chapter 11.4, Problem 69E ### Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742

#### Solutions

Chapter
Section ### Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742
Textbook Problem

# Collinear Points and Determinants(a) If three points lie on a line, what is the area of the “triangle” that they determine? Use the answer to this question, together with the determinant formula for the area of a triangle, to explain why the points ( a 1 , b 1 ) , ( a 2 , b 2 ) and ( a 3 , b 3 ) are collinear if and only if | a 1 b 1 1 a 2 b 2 1 a 3 b 3 1 | = 0 (b) Use a determinant to check whether each set of points is collinear. Graph them to verify your answer.(i) ( − 6 , 4 ) , ( 2 , 10 ) , ( 6 , 13 ) (ii) ( − 5 , 10 ) , ( 2 , 6 ) , ( 15 , − 2 )

To determine

(a)

To explain:

The statement “the points (a1,b1), (a2,b2) and (a3,b3) are collinear if and only if

|a1b11a2b21a3b31|=0”.

Explanation

Approach:

If the three points lie on the same line, the area of the triangle formed by the three points is zero.

Calculations:

If the three points lie on the same line, then the area of the triangle is zero as the third point lies on the line joining the other two points.

Thus, if the area of the triangle is zero, the three points are collinear and if the area of the triangle is not equal to zero, then the points are not collinear.

The points (a1,b1), (a2,b2) and (a3,b3) are collinear if and only if |a1b11a2b21a3b31|=0

To determine

(b)

To check:

Whether each set of points are collinear.

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