Problem 23 - APEX Calculus

Question

In Exercises 23-26, find the point of intersection between the line and the plane. line: \(\langle 5,1,-1\rangle+t\langle 2,2,1\rangle\) plane: \(5 x-y-z=-3\)

Step-by-Step Solution

Answer

The point of intersection is \((-3, -7, -5)\).

Step 1: Parameterize the line

The line can be expressed with the parametric equations derived from its vector form: \( x = 5 + 2t \), \( y = 1 + 2t \), \( z = -1 + t \).

Step 2: Substitute the line's parametric equations into the plane's equation

Substitute the parametric equations for \(x\), \(y\), and \(z\) into the equation of the plane \(5x - y - z = -3\). This means replacing the variables with the expressions from the parametric form: \[ 5(5 + 2t) - (1 + 2t) - (-1 + t) = -3 \].

Step 3: Simplify the equation

Expand and simplify the equation: \[ 25 + 10t - 1 - 2t + 1 - t = -3 \]. Collect like terms: \[ 25 + 7t = -3 \].

Step 4: Solve for the parameter t

Rearrange the equation to find \(t\): \[ 7t = -3 - 25 \], which simplifies to \[ 7t = -28 \]. Therefore, solving for \(t\), \[ t = -4 \].

Step 5: Find the intersection point

Substitute \( t = -4 \) back into the parametric equations to find the coordinates of the intersection point: \( x = 5 + 2(-4) = -3 \), \( y = 1 + 2(-4) = -7 \), \( z = -1 + (-4) = -5 \). Thus, the intersection point is \((-3, -7, -5)\).

Key Concepts

Parametric Equations

Parametric equations are a way of expressing a line in three-dimensional space by using parameters, usually denoted by a letter like \( t \). For the given line, we start with its vector form: \( \langle 5, 1, -1 \rangle + t \langle 2, 2, 1 \rangle \). This form allows us to express each coordinate in terms of the parameter \( t \).

For example:

• \( x = 5 + 2t \)
• \( y = 1 + 2t \)
• \( z = -1 + t \)

These equations show how changing the value of \( t \) moves us along the line. By substituting different \( t \) values, we can find different points on that line, which is key in determining the line's course through space.

Plane Equation

A plane in three-dimensional space can be described by a linear equation, such as the plane equation \( 5x - y - z = -3 \). This equation incorporates the variables \( x \), \( y \), and \( z \), representing coordinates of any point on the plane. This type of equation, often written as \( ax + by + cz = d \), where \( a \), \( b \), and \( c \) are coefficients, defines a flat, two-dimensional surface in a three-dimensional space.

A plane equation lets us determine whether a particular point or a line is on the plane. For solving the intersection of a line with this plane, we substitute the parametric equations of the line into the plane equation. This substitution shows us where the line 'cuts through' the plane, by identifying specific values for \( t \) that result in points that lie exactly on the plane.

System of Linear Equations

When finding the intersection of lines and planes, we often encounter a system of linear equations, which arise from substituting expressions for \( x \), \( y \), and \( z \) into the plane equation. In our example, the substituted form was \( 5(5 + 2t) - (1 + 2t) - (-1 + t) = -3 \). Solving this system involves simplifying and rearranging terms to find the parameter \( t \). The simplification process involves steps such as expanding brackets, collecting like terms, and isolating \( t \).

Systematic solving reveals that the value of \( t \) is \(-4\), providing the specific point on the line that intersects with the plane. So, linear equation systems help us transform multiple expressions into a manageable solution.

Vector Form

The vector form of a line is fundamental in visualizing and defining its behavior in space. In our exercise, the line’s vector form \( \langle 5, 1, -1 \rangle + t \langle 2, 2, 1 \rangle \) consists of two components:

• The starting point or position vector: \( \langle 5, 1, -1 \rangle \)
• Direction vector scaled by \( t \): \( t \langle 2, 2, 1 \rangle \)

The position vector specifies where the line starts, while the direction vector indicates the line's direction and slope relative to each coordinate axis. The direction vector determines how far and in which direction the line extends when \( t \) changes.

Knowing a line's vector form is crucial for establishing its parametric equations and identifying its interactions, like intersections or parallelism, with planes or other lines in the same space.