91Ó°ÊÓ

A paraboloid is a three-dimensional surface which can be thought of as a generalization of a parabola. It is defined by a quadratic equation in two variables, typically expressed as \(z = ax^2 + by^2\). This specific shape can be either an elliptic paraboloid, which opens upward like a dish, or a hyperbolic paraboloid, which resembles a saddle.
This exercise involves an elliptic paraboloid given by the equation \(z = x^2 + 3y^2\). Here, the coefficients of \(x^2\) and \(y^2\) determine the stretching of the paraboloid along the \(x\)-axis and \(y\)-axis, respectively. The paraboloid intersects a plane to form a curve \(C\), and it is on this curve that the tangent line we need to find will be located.

Parametric equations are used to express geometric locations as functions of one or more variables, called parameters. Instead of representing a curve in Cartesian coordinates \((x, y, z)\), parametric equations use variables like \(t\) to describe the position coordinates in terms of one or more parametric equations:

• \(x(t)\) to describe the \(x\)-coordinate,
• \(y(t)\) for the \(y\)-coordinate,
• \(z(t)\) for the \(z\)-coordinate.

This exercise results in parametric equations for the tangent line, such as \(x(t) = 2 + ct, y(t) = 1 - ct, z(t) = 7 + ct\). These expressions effectively describe the line in three-dimensional space as \(t\), the parameter, varies.

The gradient vector is a critical component in multivariable calculus and vector analysis. It represents the direction and rate of fastest increase of a function at a given point. For a function \(f(x, y, z)\), the gradient (denoted as \(abla f\)) is composed of the partial derivatives of \(f\) with respect to each variable:

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

In this exercise, the gradient of the paraboloid \(z = x^2 + 3y^2\) at the point \((2,1,7)\) is \(\langle 4, 6, -1\rangle\). This vector is normal to the tangent plane of the paraboloid at the given point, meaning it is perpendicular to any directional vector lying in that plane.

A direction vector provides the direction in which a line or curve extends. In geometry and vector calculus, the direction vector is essential for constructing lines in space. It can be denoted as \(\langle a, b, c \rangle\) and indicates how much movement occurs in the \(x\), \(y\), and \(z\) directions respectively.
In this exercise, we identify the direction vector for the tangent line by ensuring it is perpendicular to both normal vectors of the intersecting planes, leading to equations like \(a + b - c = 0\) and \(4a + 6b - c = 0\). Solving these equations gives us a direction vector \(\langle c, -c, c \rangle\). This property ensures the tangent line lies along the curve formed by the intersection of the plane and the paraboloid.