Question

Find the point (if it exists) at which the following planes and lines intersect. $$z=4 ; \mathbf{r}(t)=\langle 2 t+1,-t+4, t-6\rangle$$

Short answer

Question: Find the point of intersection between the plane z = 4 and the parametric line equation r(t) = ⟨2t + 1, -t + 4, t - 6⟩. Answer: The point of intersection is (21, -6, 4).

Step-by-step solution

Identify the given equations

We have a plane equation: $$z = 4$$ And a parametric line equation: $$\mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle = \langle 2t + 1, -t + 4, t - 6\rangle$$

Substitute the line equation into the plane equation

Replace the \(z\) in the plane equation with the \(z(t)\) from the line equation: $$t - 6 = 4$$

Solve for t

Now, solve for \(t\): $$t - 6 = 4 \Rightarrow t = 10$$

Substitute the value of t into the line equation

Now that we have the value of \(t\), plug it into the line equation to find the intersection point: $$x(10) = 2(10) + 1 = 21$$ $$y(10) = -(10) + 4 = -6$$ $$z(10) = 10 - 6 = 4$$

Write the intersection point

The point of intersection between the given plane and line is: $$(21, -6, 4)$$

Key concepts

Parametric Equations

Parametric equations are an essential tool in mathematics, particularly useful when describing the motion of an object in space. They allow us to define each coordinate position as a function of a single parameter, typically time. In our problem, the parametric line equation is given by
\( \mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle = \langle 2t + 1, -t + 4, t - 6\rangle \).
This equation describes a line where the positions of the x, y, and z coordinates are functions of the parameter t. When we want to find an intersection with a plane, as in this exercise, we can evaluate these equations for specific values of t, providing us with the specific points on the line at any given moment. Through parametric equations, complex motions and intersections are broken down into simpler one-dimensional problems for ease of understanding and resolution.

Plane Equations

The equation of a plane in three-dimensional space can be represented in several forms, with the most straightforward being the form \( Ax + By + Cz + D = 0 \). In the context of our exercise, the plane equation is given by
\( z = 4 \).
This specific form tells us that for any point \( (x, y, z) \) on the plane, the z-coordinate will always be 4, regardless of the values of x and y. This is akin to an infinite flat surface that intersects the z-axis at the point where z equals 4. To locate a point of intersection with a line, we seek the specific x and y values that, along with this constant z-value, satisfy both the plane and line equations simultaneously.

Solving Systems of Equations

Solving systems of equations is the process of finding the values for the variables that simultaneously satisfy all equations in the system. In our case, the system consists of a single plane equation and a parametric line equation. We look for a specific value of the parameter t that allows the x(t), y(t), and z(t) from the parametric equation to satisfy the plane equation
\( z = 4 \).
By replacing the z-coordinate from the parametric equation into the plane equation, we form a new equation with just one variable, t. Solving for t gives us the precise moment when the moving point described by the parametric equation intersects the plane. Once we have the value of t, we can then find the corresponding x and y values to pinpoint the exact location of the intersection.

Calculus

Calculus, the mathematical study of change, provides tools for analyzing functions and rates at which quantities change. Even though calculus is not directly employed in this problem, its fundamental concepts underlie much of what we do when dealing with equations of lines and planes. For example, the concept of a derivative, a cornerstone of calculus, gives us the slope of a function at any point. In the context of parametric equations, derivatives with respect to the parameter can help us understand how the position of a point on a line changes over time. This understanding can help us solve more complicated intersection problems where, instead of a static plane, we might have a surface that is changing shape or size—contexts where calculus would become indispensable.