Question

In Exercises \(81-84,\) find the point(s) of intersection (if any) of the plane and the line. Also determine whether the line lies in the plane. $$ 2 x-2 y+z=12, \quad x-\frac{1}{2}=\frac{y+(3 / 2)}{-1}=\frac{z+1}{2} $$

Short answer

First, convert the line equation into parametric form. Then, using the parametric equations, substitute these equations into the plane equation, and solve for the parameter 't'. Once 't' is found, substitute its value back into the line's parametric equations to find the point of intersection of the line and the plane. Verify finally if there exists more than one solution for 't', thus proving the line lies in the plane or not.

Step-by-step solution

Obtain the parametric equations of the line

The line equation given in symmetric form is: \(x-(1/2)=\frac{y+(3/2)}{-1}=\frac{z+1}{2}\). Equality of ratios allows writing each component as equal to 't', a parameter. Hence, the parametric equations would be: \(x = t + 1/2\), \(y = -t - 3/2\) and \(z = 2t - 1\).

Substitute the parameters in the equation of the plane

Substitute the parametric equations of the line into the plane equation given as: \(2x - 2y + z = 12\). This would result to: \(2(t + 1/2) - 2(-t - 3/2) + (2t - 1) = 12\). Solving this equation gives the parameter definition, 't', of the line that intersects the plane.

Verify the solution

If 't' is a real number, the line intersects the plane at one specific point. To find this point, substitute 't' back into the line parametric equations. If there is no valid solution for 't', the line does not intersect with the plane. At the end check if there are multiple solutions for 't', in which case, the line lies entirely in the plane.

Key concepts

Parametric Equations

Parametric equations are a way of expressing a mathematical relationship in terms of one or more independent variables, known as parameters. For example, a line in three-dimensional space can be described using a parameter \(t\), with separate equations for each coordinate axis:
\[x = x_{0} + at, \quad y = y_{0} + bt, \quad z = z_{0} + ct\]
These equations define the line in terms of \(t\), where \((x_{0}, y_{0}, z_{0})\) is a point on the line, and \((a, b, c)\) are the direction ratios, corresponding to the line's direction in space. Parametric equations are particularly useful because they can describe curves and surfaces in addition to lines. In the given exercise, we convert the symmetrical form of a line into its parametric form to interface with the plane equation more efficiently.

Understanding the Parameters

In the given example, each coordinate, \(x, y,\) and \(z\), is expressed in terms of the parameter \(t\). This allows us to analyze the position of any point along the line simply by substituting the value of \(t\). The ability to plug in different values of \(t\) into the parametric equations makes it easy to determine whether and where the line intersects with other geometric forms, such as planes.

Plane Equation

A plane equation in three-dimensional space provides a way to describe a flat, infinite surface through an algebraic expression. Typically, it is expressed in the form:
\[Ax + By + Cz + D = 0\]
where \(A\), \(B\), and \(C\) represent the normal vector to the plane and \(D\) is a constant term. The normal vector gives us the direction perpendicular to every point on the plane. To determine the relationship between a plane and a line, the parametric equations of the line can be substituted into the plane's equation.

Application in Intersection Problems

For an intersection problem like in our exercise, we are substituting the values from the line's parametric equations directly into the plane's equation. If the resultant equation has a valid solution, it means that the line intersects the plane at the point corresponding to that particular value of \(t\). It’s essential to note that the normal vector plays a crucial role in understanding the orientation of the plane, which can affect the intersection with the line.

Line Intersection

Line intersection with a plane occurs at the point where the line passes through the plane in three-dimensional space. This concept is significant as it helps to understand how different geometrical entities interact with each other. To find the intersection, we replace the 'x', 'y', and 'z' in the plane equation with the parametric equations of the line. Solving for the parameter gives us the intersection point (if any).

Checking for Line Intersections

After substitution and solving the equation, if \(t\) yields a real number, it indicates a single point of intersection. If no solution exists for \(t\), the line and plane do not intersect. Crucially, if the substitution leads to an identity (true for all values of \(t\)), it means the line lies in the plane, as all points on the line satisfy the plane equation. In the provided exercise, once we know the parameter \(t\), we would plug it back into the parametric equations of the line to find the precise coordinates of the intersection point, hence demonstrating a practical application of intersection theory in geometry.