91Ó°ÊÓ

A normal line to a surface is a line that is perpendicular to the tangent plane at a given point. Understanding the normal line is crucial because it helps define the orientation of the tangent plane in three-dimensional space.

In this exercise, the normal line is derived at the point on the surface determined by the equation \(2z - x^2 = 0\). Given the point \( P_0(2, 0, 2) \), we found the gradient vector at this point, which serves as the direction vector for the normal line. The gradient vector calculated is \((-4, 0, 2)\).

To formulate the normal line, we use the parametric equations which introduce a variable \(t\) that allows for movement along the line:

• \( x = 2 - 4t \)
• \( y = 0 \)
• \( z = 2 + 2t \)

This set of equations fully describes the normal line extending from the point \(P_0\) along the surface.

The gradient vector is an essential tool in multivariable calculus, representing the direction of the steepest ascent of a function. When dealing with surfaces, it acts as a normal vector to the surface at a given point.

For the surface equation \(2z - x^2 = 0\), the function to consider is \( f(x, y, z) = 2z - x^2 \). The gradient vector \( abla f \) for this function is calculated by finding the partial derivatives with respect to each variable:

• \( \frac{\partial f}{\partial x} = -2x \)
• \( \frac{\partial f}{\partial y} = 0 \)
• \( \frac{\partial f}{\partial z} = 2 \)

Substituting the coordinates of the point \(P_0(2, 0, 2)\) into the gradient vector formula gives \((-4, 0, 2)\). This vector is used in determining both the tangent plane and the normal line equations.

A surface equation describes a surface in three-dimensional space. It encapsulates the relationship between the coordinates \((x, y, z)\) that make up points on the surface.

In this particular exercise, the surface equation is given by \(2z - x^2 = 0\). This equation defines a curved surface known as a quadratic surface. The structure indicates that for every value of \(z\), the values \(x\) that satisfy the equation describe the shape of the surface.

With the point \( P_0(2, 0, 2) \) identified, we explored how this surface behaves near this particular point, leading us to the tangent plane and normal line. The surface equation is the foundational component that guides these calculations and helps visualize the geometric properties.

Parametric equations are ways of expressing points or paths along a line or curve using a parameter, often denoted as \(t\). Instead of expressing \(y\) directly in terms of \(x\), parametric equations use a third variable to define \(x\), \(y\), and \(z\) separately.

For the normal line in this exercise, we used parametric equations to describe the coordinates of any point on the line extending from \(P_0(2, 0, 2)\) in the direction given by the gradient vector \((-4, 0, 2)\). The parametric form allowed us to define:

• \(x = 2 - 4t\) - the x-component
• \(y = 0\) - the y-component, which remains constant
• \(z = 2 + 2t\) - the z-component

These equations result in a more flexible, continuous representation of the normal line, making it easier to map out or visualize in applications that require detailed analysis of geometric shapes. They are crucial for both theoretical insights and practical computations in mathematics.