Chapter 10: Q 21. (page 845)

Find the equations of the planes determined by the given conditions.
The plane contains the origin and is normal to the vector $(4, - 1, 5)$

Short Answer

Expert verified

The equation of the plane that is determined by the given conditions is $4 x - 1 y + 5 z = 0$

Step by step solution

Given information

The plane that is normal to the vector $(4, - 1, 5)$ and contains the origin.

Calculation

Consider the plane that is normal to the vector $⟨ 4, - 1, 5 ⟩$ and contains the origin.

The goal is to determine the plane equation that is determined by the given conditions.

The equation of the plane with $r_{0}$ a point that lies in the plane and $n$ a vector normal to the plane is given by:

$n \cdot (r - r_{0}) = 0$

The normal vector is:

$n = ⟨ 4, - 1, 5 ⟩$

The point $r_{0}$ is:

$r_{0} = (0, 0, 0)$

Calculation

The plane that contains the origin and is normal to the vector $⟨ 4, - 1, 5 ⟩$ has the equation:

$⟨ 4, - 1, 5 ⟩ \cdot (⟨ x, y, z ⟩ - ⟨ 0, 0, 0 ⟩) = 0 (Substitution) ⟨ 4, - 1, 5 ⟩ \cdot ⟨ x, y, z ⟩ = 0 (Simplify) 4 x - 1 y + 5 z = 0$

Thus, the equation of the plane that is determined by the given conditions is $4 x - 1 y + 5 z = 0$

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Find the equations of the planes determined by the given conditions.
The plane contains the origin and is normal to the vector $(4, - 1, 5)$

Short Answer

Step by step solution

Given information

Calculation

Calculation

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Find the equations of the planes determined by the given conditions.The plane contains the origin and is normal to the vector4,−1,5

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

Calculus

Probability and Statistics

Pure Maths

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Theoretical and Mathematical Physics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Find the equations of the planes determined by the given conditions.
The plane contains the origin and is normal to the vector $(4, - 1, 5)$