Q 57. Prove that the planes determined... [FREE SOLUTION]

Chapter 10: Q 57. (page 846)

Prove that the planes determined by the equations $a x + b y + c z = d$ and $伪 x + 尾 y + 纬 z = 未$ are perpendicular if and only if $伪 x + 尾 y + 纬 z = 未$

Short Answer

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The planes determined by the equations $a x + b y + c z = d$ and $伪 x + 尾 y + 纬 z = 未$ are perpendicular if and only if $a 伪 + b 尾 + c 纬 = 0$

Step by step solution

Given information

The planes determined by the equations $a x + b y + c z = d$ and $伪 x + 尾 y + 纬 z = 未$

Calculation

The goal is to show that planes deduced from provided equations are perpendicular if and only if $a 伪 + b 尾 + c 纬 = 0$

The normal vector of the equation $a x + b y + c z = d$ is $N_{1} = 鉄� a, b, c 鉄�$ and of the equation $伪 x + 尾 y + 纬 z = 未$ is $N_{2} = 鉄� 伪, 尾, 纬鉄�$

Calculation

Only if and only if the planes' normal vectors are orthogonal are they considered orthogonal.

The normal vectors are orthogonal if and only if $N_{1} 路 N_{2} = 0$

The orthogonal condition $N_{1} 路 N_{2} = 0$ gives:

鉄� a, b, c 鉄� 路 鉄� 伪, 尾, 纬 鉄� = 0 a 伪 + b 尾 + c 纬 = 0 (Dot Product)

Thus, the planes determined by the equations $a x + b y + c z = d$ and $伪 x + 尾 y + 纬 z = 未$ are perpendicular if and only if $a 伪 + b 尾 + c 纬 = 0$

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91影视

Prove that the planes determined by the equations $a x + b y + c z = d$ and $伪 x + 尾 y + 纬 z = 未$ are perpendicular if and only if $伪 x + 尾 y + 纬 z = 未$

Short Answer

Step by step solution

Given information

Calculation

Calculation

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91影视

Prove that the planes determined by the equations ax+by+cz=d and 伪x+尾y+纬z=未 are perpendicular if and only if 伪x+尾y+纬z=未

Short Answer

Step by step solution

Given information

Calculation

Calculation

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Mechanics Maths

Calculus

Applied Mathematics

Theoretical and Mathematical Physics

Pure Maths

Study anywhere. Anytime. Across all devices.

Prove that the planes determined by the equations $a x + b y + c z = d$ and $伪 x + 尾 y + 纬 z = 未$ are perpendicular if and only if $伪 x + 尾 y + 纬 z = 未$