Chapter 10: Q 45. (page 846)

Use your answers from Exercise 14 to find the angle between the indicated planes in Exercises 44 and 45.
$- x + 7 y - 2 z = 5$ and $3 x + 5 y - 4 z = 2$

Short Answer

Expert verified

The angle between two planes is $θ = \cos^{- 1} (\frac{4 \sqrt{3}}{9})$

Step by step solution

Given information

The two planes $- x + 7 y - 2 z = 5$ and $3 x + 5 y - 4 z = 2$

Calculation

The goal is to determine the angle between the two planes shown.

The angle formed by two planes can be calculated as follows:

θ = \cos^{- 1} (\frac{N_{1} \cdot N_{2}}{||N_{1}|| ||N_{2}||})

Calculation

The normal vector of the plane $- x + 7 y - 2 z = 5$ is $N_{1} = ⟨ - 1, 7, - 2 ⟩$ and the normal vector of the plane $3 x + 5 y - 4 z = 2$ is $N_{2} = ⟨ 3, 5, - 4 ⟩$

The angle between the two planes is:

$θ = \cos^{- 1} (\frac{⟨ - 1, 7, - 2 ⟩ \cdot ⟨ 3, 5, - 4 ⟩}{‖ (- 1, 7, - 2 ⟩ ‖ ‖ (3, 5, - 4 ⟩ ‖})$ (Substitution)

$= \cos^{- 1} (\frac{- 1 (3) + 7 (5) - 2 (- 4)}{\sqrt{1 + 49 + 4} \sqrt{9 + 25 + 16}}) (Dot Product)$

$= \cos^{- 1} (\frac{40}{\sqrt{54} \sqrt{50}})$ (Simplify)

$= \cos^{- 1} (\frac{40}{3 \sqrt{6} \times 5 \sqrt{2}})$

$= \cos^{- 1} (\frac{4}{3 \sqrt{3}})$

$= \cos^{- 1} (\frac{4 \sqrt{3}}{9})$ (Rationalize)

The angle between two planes is $θ = \cos^{- 1} (\frac{4 \sqrt{3}}{9})$

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Use your answers from Exercise 14 to find the angle between the indicated planes in Exercises 44 and 45.
$- x + 7 y - 2 z = 5$ and $3 x + 5 y - 4 z = 2$

Short Answer

Step by step solution

Given information

Calculation

Calculation

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Use your answers from Exercise 14 to find the angle between the indicated planes in Exercises 44 and 45. −x+7y−2z=5 and 3x+5y−4z=2

Short Answer

Step by step solution

Given information

Calculation

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Statistics

Logic and Functions

Theoretical and Mathematical Physics

Discrete Mathematics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Use your answers from Exercise 14 to find the angle between the indicated planes in Exercises 44 and 45.
$- x + 7 y - 2 z = 5$ and $3 x + 5 y - 4 z = 2$