91Ó°ÊÓ

Parametric equations are a way of representing a line or curve in terms of parameters, usually using one or more independent variables. In the context of a tangent line, parametric equations are useful because they allow us to describe the line in terms of a point and a direction, capturing both the position and the orientation.
To set up parametric equations for a line, we need:

• A point through which the line passes.
• A direction vector, which gives the orientation of the line.

The parametric equations take the form:

• \[ x = x_0 + at \]
• \[ y = y_0 + bt \]
• \[ z = z_0 + ct \]

where \((x_0, y_0, z_0)\) is the point, \((a, b, c)\) is the direction vector, and \(t\) is the parameter that we can vary.
In our problem, the point is given as \(\left( \frac{1}{2}, 1, \frac{1}{2} \right)\) and the direction vector as \((-1, 0, 1)\). So the parametric equations are:

• \[ x = \frac{1}{2} - t \]
• \[ y = 1 \]
• \[ z = \frac{1}{2} + t \]

A gradient vector is a vector that represents the rate and direction of change in a function. It is composed of the partial derivatives of a function with respect to each variable. Gradients are essential in calculus, particularly when trying to find tangent lines or optimizing functions.
For a function \(f(x, y, z)\), the gradient \(abla f\) is expressed as:

• \[ abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) \]

In our problem, we have two surfaces:

• Surface 1: \(x + y^2 + z = 2\) with \(abla f = (1, 2y, 1)\). At the point \((\frac{1}{2}, 1, \frac{1}{2})\), it simplifies to \((1, 2, 1)\).
• Surface 2: \(y = 1\) with \(abla g = (0, 1, 0)\), indicating that the surface is constant with respect to \(x\) and \(z\), so it doesn't vary in those directions.

These gradient vectors help us find the direction vector for the tangent line by using methods involving a vector cross product.

The cross product is a vector operation that takes two vectors and produces a third vector perpendicular to both. It's used in three dimensions, and the result is critical in determining the direction of a line or a plane.
The cross product of vectors \(\mathbf{A} = (a_1, a_2, a_3)\) and \(\mathbf{B} = (b_1, b_2, b_3)\) is given by:

• \[ \mathbf{A} \times \mathbf{B} = (a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1) \]

In our problem, to find the direction vector, we use the cross product of the gradient vectors of the two surfaces.

• Gradient of Surface 1: \(abla f = (1, 2, 1)\)
• Gradient of Surface 2: \(abla g = (0, 1, 0)\)

The resulting cross product \(abla f \times abla g\) is the direction vector of the tangent line:

• \[ (-1, 0, 1) \]

A direction vector is a vector that points in the direction of a line or vector space. It represents the line's orientation and determines how the line extends from any given point.
To find the direction vector for a line tangent to the intersection of two surfaces, a common technique is to use the cross product of their respective gradient vectors.
In this exercise, we obtained:

• Gradient of Surface 1: \(abla f = (1, 2, 1)\)
• Gradient of Surface 2: \(abla g = (0, 1, 0)\)
• Direction vector through cross product: \((-1, 0, 1)\)

The cross product here provides a direction vector \((-1, 0, 1)\) perpendicular to both gradient vectors, ensuring our tangent line is correctly oriented at the point of intersection.
This direction vector helps build parametric equations, confirming the line's correct orientation by ensuring it lies on the curves that define the intersection.