Problem 170 - Calculus Volume 3

Question

Find the equation for the tangent plane to the surface at the indicated point. \(-8 x-3 y-7 z=-19, P(1,-1,2)\)

Step-by-Step Solution

Answer

The tangent plane equation is \(-8x - 3y - 7z = -19\).

Step 1: Identify the Surface Equation

The given surface is described by the equation \(-8x - 3y - 7z = -19\). This is the equation of a plane. We need to find the tangent plane at the point \(P(1, -1, 2)\), which actually lies on this plane.

Step 2: Understand Planes in Manipulation

Generally, the equation of a plane has the form \(Ax + By + Cz = D\). In this equation already given: \(-8x - 3y - 7z + 19 = 0\). For a tangent plane to this plane, it remains the same, as it is the same surface everywhere.

Step 3: Confirm the Point Lies on the Plane

Substitute \(x = 1\), \(y = -1\), and \(z = 2\) into the plane equation \(-8x - 3y - 7z = -19\) to ensure the point \(P(1, -1, 2)\) lies on the plane. Calculate: \((-8)(1) - 3(-1) - 7(2) = -8 + 3 - 14 = -19\). The point satisfies the equation and lies on the given plane.

Step 4: Conclusion on Tangent Plane

Since any plane is tangent to itself at every point, the equation of the tangent plane at the point \(P(1, -1, 2)\) is simply \(-8x - 3y - 7z = -19\). There is no need to change the coefficients or the format of the equation.

Key Concepts

Plane Equation

A plane equation represents a flat, two-dimensional surface in three-dimensional space. It usually takes the form \(Ax + By + Cz = D\), where \(A\), \(B\), and \(C\) are constants that determine the orientation of the plane, and \(D\) is a constant that shifts the plane from the origin.For the equation \(-8x - 3y - 7z = -19\), the constants \(-8\), \(-3\), and \(-7\) are essential because they determine the direction of the plane. Specifically, these numbers form a vector, called the normal vector, \(\mathbf{n} = \langle -8, -3, -7 \rangle\). This vector is crucial because it is perpendicular to the plane.

• This equation provides a simple way to describe a plane using algebraic terms.
• The coefficients correspond directly to the plane's orientation in space.
• Any point \((x, y, z)\) that satisfies the equation lies on the plane.

Surface Equation

The surface equation describes the flat plane on which we are working, and a plane is the simplest type of surface in three-dimensional space. In this exercise, the surface equation refers to the entire expression \(-8x - 3y - 7z = -19\). Surfaces extend indefinitely unless otherwise bounded and can have different orientations.For tangent planes, it's important to understand that each point on a surface has a tangent plane. However, since the given problem specifically deals with a flat plane, the entire surface and all its tangent planes look the same.

• Tangent planes provide immediate approximations of the surface at a local point.
• Tangents and normals are key concepts when dealing with curves and surfaces.
• Understanding the nature of the surface helps in knowing how the tangent plane behavior aligns with it.

Point on Plane

Determining if a point lies on a plane involves checking whether its coordinates satisfy the plane's equation. For instance, to confirm if point \(P(1, -1, 2)\) is on the plane given by \(-8x - 3y - 7z = -19\), substitute the point into the equation: \[-8(1) - 3(-1) - 7(2) = -8 + 3 - 14 = -19.\]Since both sides of the equation balance, the point \(P(1, -1, 2)\) does indeed lie on the surface of the plane.

• Check by substituting point coordinates into the equation.
• If both sides equal, the point is confirmed to lie on the plane.
• For tangent plane exercises, this verification is crucial to confirm our working plane.