91Ó°ÊÓ

A gradient vector is crucial when finding the equation of a tangent plane to a surface defined by a scalar function. In simple terms, the gradient vector provides the direction of the steepest ascent for a multivariable function. For the ellipsoid given by the equation \(x^2 + y^2 + 4z^2 = 12\), the gradient vector is found by computing the partial derivatives of each variable in the function:

• \(\frac{\partial f}{\partial x} = 2x\)
• \(\frac{\partial f}{\partial y} = 2y\)
• \(\frac{\partial f}{\partial z} = 8z\)

Evaluating these at point \((2, 2, 1)\), we determine the gradient vector as \((4, 4, 8)\). This vector not only helps in defining the tangent plane but also plays a key role in constructing the equation of any normal line to the surface at a specific point.

Parametric equations are helpful in expressing lines and curves by setting each spatial variable as a function of one or more parameters. In the context of the normal line to an ellipsoid, the parametric equations allow us to describe the line in terms of a single variable, often denoted by \(t\). For instance, given a point \((2, 2, 1)\) on our ellipsoid and a direction vector \((4, 4, 8)\), the parametric equations of the normal line are derived as follows:

• \(x = 2 + 4t\)
• \(y = 2 + 4t\)
• \(z = 1 + 8t\)

Here, \(t\) varies over all real numbers, and each equation represents one dimension of space. As \(t\) changes, the equations describe a line traveling through the point \((2, 2, 1)\) along the direction of the gradient vector \((4, 4, 8)\) that is normal to the surface.

Determining the angle between two planes involves understanding their normal vectors. For two planes with normal vectors \(\mathbf{n}_1\) and \(\mathbf{n}_2\), the cosine of the angle \(\theta\) between them is derived from the dot product of the vectors. The formula goes:\[\cos\theta = \frac{\mathbf{n}_1 \cdot \mathbf{n}_2}{||\mathbf{n}_1|| \cdot ||\mathbf{n}_2||}\]In our example, the normal vector to the tangent plane at \((2, 2, 1)\) is \((4, 4, 8)\) and for the \(xy\)-plane, it is \((0, 0, 1)\). The dot product is 8, and their magnitudes are \(4\sqrt{6}\) and 1 respectively. Thus, the acute angle \(\theta\) between the planes is:\[\theta = \cos^{-1}\left( \frac{2}{\sqrt{6}} \right)\]This angle measures how steeply the tangent plane to the ellipsoid inclines relative to the horizontal \(xy\)-plane.

A normal line to a surface at a given point is perpendicular to the tangent plane at that point. Constructing a normal line can be approached similarly to parametrizing a line, where we use a point on the surface and the vector normal to the surface at that point (from the gradient). For the ellipsoid in our case, where the point is \((2, 2, 1)\) and the normal vector \((4, 4, 8)\), the normal line is defined by parametric equations:

• \(x = 2 + 4t\)
• \(y = 2 + 4t\)
• \(z = 1 + 8t\)

These are created by stepping from the point \((2, 2, 1)\) in the direction \((4, 4, 8)\). This method ensures the line remains perpendicular to the tangent plane, thus maintaining the geometric definition of a normal line.