Explanation Given: The lines L 1 and L 2 have slopes m 1 and m 2 as shown in the figure below: Formula Used: tan ( α − β ) = tan α − tan β 1 + tan α tan β Now consider the figure given below: Here, ∠ 1 and ∠ ϕ are vertically opposite angles. So, they are equal. That is, ∠ 1 = ∠ ϕ Also, it is known that the exterior angle of a triangle is equal to the sum of its interior opposite angles. Thus, in △ A B C , ∠ α 2 = ∠ α 1 + ∠ 1 = ∠ α 1 + ∠ ϕ This gives that, ∠ ϕ = ∠ α 2 − ∠ α 1 Thus, tan ϕ = tan ( α 2 − α 1 ) = tan α 2 − tan α 1 1 + tan α 2 tan α 1 But m 1 = tan α 1 is the slope of the line L 1 and m 2 = tan α 2 is the slope of the line L 2 . Hence, tan ϕ = m 2 − m 1 1 + m 1 m 2 Now, suppose that slopes m 1 and m 2 are negative reciprocals of each other

Calculus: Special Edition: Chapters 1-5 (w/ WebAssign)

6th Edition

ISBN: 9781524908102

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

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Chapter 1.3, Problem 58PS

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To show: tanϕ=m2−m11+m1m2 . Also show that two lines have slopes that are negative reciprocals of each other if and only if they are perpendicular.

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Chapter 1.3, Problem 57PS

Chapter 1.3, Problem 59PS

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Chapter 1 Solutions

Calculus: Special Edition: Chapters 1-5 (w/ WebAssign)

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tan ϕ = m 2 − m 1 1 + m 1 m 2 . Also show that two lines have slopes that are negative reciprocals of each other if and only if they are perpendicular.

Solution Summary: The author explains that two lines have slopes that are negative reciprocals of each other if they are perpendicular.

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To show: tanϕ=m2−m11+m1m2 . Also show that two lines have slopes that are negative reciprocals of each other if and only if they are perpendicular.

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Chapter 1 Solutions