Chapter 10: Q 40. (page 846)

Show that the planes given by $2 x - 4 y - 3 z = 5$ and $- 4 x + 8 y + 6 z = 1$ are parallel, and find the distance between the planes.

Short Answer

Expert verified

The distance between parallel to the plane is $\frac{11 \sqrt{29}}{58}$

Step by step solution

Given information

The planes $2 x - 4 y - 3 z = 5$ and $- 4 x + 8 y + 6 z = 1$

Calculation

The goal is to calculate the distance between a specified point and the plane. The distance between a point and the plane is calculated as follows:

distance = \frac{|N \cdot \bar{R_{1} R_{2}}|}{‖ N ‖}

The normal vector of the plane $2 x - 4 y - 3 z = 5$ is:

N_{1} = ⟨ 2, - 4, - 3 ⟩

The normal vector of the plane $- 4 x + 8 y + 6 z = 1$ is:

N_{2} = ⟨ - 4, 8, 6 ⟩

The normal vectors $N_{1} = ⟨ 2, - 4, - 3 ⟩$ and $N_{2} = ⟨ - 4, 8, 6 ⟩$ are scalar multiple of each other. Therefore, the planes are parallel.

Calculation

The point $R_{1} = (0, 0, - \frac{5}{3})$ is lies on the plane $2 x - 4 y - 3 z = 5$ and the point $R_{2} = (0, 0, \frac{1}{6})$ lies on the plane $- 4 x + 8 y + 6 z = 1$

The vector $\vec{R_{1} R_{2}}$ is given by:

$\vec{R_{1} R_{2}} = <0 - 0, 0 - 0, \frac{1}{6} + \frac{5}{3}> \vec{R_{1} R_{2}} = <0, 0, \frac{11}{6}>$

The distance from $P$ to the plane is:

distance = \frac{|N \cdot \vec{R_{1} R_{2}}|}{‖ N ‖} = \frac{|⟨ 2, - 4, - 3 ⟩ \cdot <0, 0, \frac{11}{6}>|}{‖ ⟨ 2, - 4, - 3 ⟩ ∣} (Substitution) = \frac{|0 + 0 - 3 \times \frac{11}{6}|}{\sqrt{4 + 16 + 9}} (Dot product) = \frac{|- \frac{11}{2}|}{\sqrt{29}} = \frac{11}{2 \sqrt{29}} (Simplify) = \frac{11 \sqrt{29}}{58}

Therefore, the distance between parallel to the plane is $\frac{11 \sqrt{29}}{58}$

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Show that the planes given by $2 x - 4 y - 3 z = 5$ and $- 4 x + 8 y + 6 z = 1$ are parallel, and find the distance between the planes.

Short Answer

Step by step solution

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Show that the planes given by 2x−4y−3z=5 and −4x+8y+6z=1 are parallel, and find the distance between the planes.

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

Applied Mathematics

Theoretical and Mathematical Physics

Mechanics Maths

Decision Maths

Geometry

Pure Maths

Study anywhere. Anytime. Across all devices.

Show that the planes given by $2 x - 4 y - 3 z = 5$ and $- 4 x + 8 y + 6 z = 1$ are parallel, and find the distance between the planes.