Question

Which of the points $A(-4,0,-1), B(3,1,-5)$, and $C(2,4,6)$ is closest to the $y z$-plane? Which point lies in the $x z$-plane?

   Which of the points $A(-4,0,-1), B(3,1,-5)$, and $C(2,4,6)$ is closest to the $y z$-plane? Which point lies in the $x z$-plane?

Calculus

James Stewart 7th Edition

Chapter 12, Problem 3

Instant Answer

Solved by Expert Howard Francis

05/29/2020

Step 1

The distance of a point \((x, y, z)\) from the \( yz \)-plane is given by the absolute value of its \( x \)-coordinate, \(|x|\). Show more…

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Which of the points $A(-4,0,-1), B(3,1,-5)$, and $C(2,4,6)$ is closest to the $y z$-plane? Which point lies in the $x z$-plane?

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Key Concepts

Coordinate Planes

Coordinate planes are the fundamental flat surfaces defined by setting one of the coordinate variables to zero in a three?dimensional space. For instance, the yz-plane is where x equals 0, the xz-plane is where y equals 0, and the xy-plane is where z equals 0. Understanding these planes helps in determining the relative positions of points in space with respect to these defined surfaces.

Distance from a Point to a Plane

The distance from a point to a plane, especially in the context of coordinate planes, can often be determined by the absolute value of the coordinate that is not represented in the plane. For example, the distance from a point to the yz-plane is simply the absolute value of its x-coordinate, as the yz-plane is defined by x = 0. This concept is crucial for comparing distances and identifying which point is closest to a given plane.

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