Problem 100 - Calculus Early Transcendentals

Question

Show that the plane \(a x+b y+c z=d\) and the line \(\mathbf{r}(t)=\mathbf{r}_{0}+\mathbf{v} t,\) not in the plane, have no points of intersection if and only if \(\mathbf{v} \cdot\langle a, b, c\rangle=0 .\) Give a geometric explanation of the result.

Step-by-Step Solution

Answer

And what does this mean geometrically?** A line does not intersect a plane when the direction vector of the line, \(\mathbf{v}\), is orthogonal to the normal vector of the plane, \(\langle a, b, c \rangle\), which can be represented as \(\mathbf{v} \cdot \langle a, b, c \rangle = 0\). Geometrically, this means the line is parallel to the plane, as it never intersects the plane and its direction vector is perpendicular to the plane's normal vector.

Step 1: Write Equations

The given equation of the plane is: $$ a x + b y + c z = d. $$ And the given vector equation of the line is: $$ \mathbf{r}(t) = \mathbf{r}_{0} + \mathbf{v}t, $$ where \(\mathbf{r}_{0}\) is the position vector of a point on the line and \(\mathbf{v}\) is the direction vector of the line.

Step 2: Check Intersection Condition

To check if the line intersects the plane, we need to substitute the coordinates of the vector equation \(\mathbf{r}(t)\) into the equation of the plane. Denote the coordinates of \(\mathbf{r}(t)\) as \((x, y, z)\). Then we get: $$ a(x_{0} + vt_{x}) + b(y_{0} + vt_{y}) + c(z_{0} + vt_{z}) = d, $$ where \(\mathbf{r}_{0} = \langle x_{0}, y_{0}, z_{0} \rangle\), \(\mathbf{v} = \langle vt_{x}, vt_{y}, vt_{z} \rangle\) and \(t\) is a scalar parameter.

Step 3: Prove No Intersection Condition

Rearrange the equation from Step 2: $$ a vt_{x} + b vt_{y}+ c vt_{z} = d - (a x_{0} + b y_{0} + cz_{0}). $$ We know that the line and the plane have no point of intersection if and only if the equation above has no solutions for \(t\). This happens when the left-hand side and the right-hand side are two independent linear combinations. In other words, this occurs when the direction vector \(\mathbf{v}\) and the normal vector of the plane \(\langle a, b, c \rangle\) are orthogonal. That is: $$ \mathbf{v} \cdot \langle a, b, c \rangle = 0. $$

Step 4: Geometric Explanation

We have shown that the plane \(ax + by + cz = d\) and the line \(\mathbf{r}(t) = \mathbf{r}_{0} + \mathbf{v}t\) do not intersect each other if and only if the direction vector of the line \(\mathbf{v}\) is orthogonal to the normal vector of the plane \(\langle a, b, c \rangle\). Geometrically, this means that the line is parallel to the plane, since it never intersects it and its direction vector is perpendicular to the plane's normal vector. Thus, the condition \(\mathbf{v} \cdot \langle a, b, c \rangle = 0\) serves as a mathematical expression of the line being parallel to the plane without intersecting it.

Key Concepts

Plane Equation

A plane equation is a mathematical expression that represents a flat, two-dimensional surface extending infinitely in three-dimensional space. The standard form of a plane equation is given by

• \( ax + by + cz = d \)

Here, \( a, b, \) and \( c \) are the coefficients that define the orientation of the plane, while \( d \) is a constant that shifts the plane along the axis.
The vector \( \langle a, b, c \rangle \) that appears in the plane equation is called the normal vector. This vector is crucial because it is perpendicular to every line lying on the plane.
Understanding this equation is essential, as it forms the basis for solving problems involving planes in linear algebra, determining their intersections, and evaluating orthogonal relationships.

Vector Equation

The vector equation of a line is a mathematical way to describe lines in vector spaces. It is expressed as

• \( \mathbf{r}(t) = \mathbf{r}_{0} + \mathbf{v}t \)

In this equation, \( \mathbf{r}_{0} \) represents a fixed point on the line, often known as the position vector, and \( \mathbf{v} \) is the direction vector, indicating the direction in which the line extends.
The parameter \( t \) is a real number that scales the direction vector, effectively moving the point \( \mathbf{r}_{0} \) along the line.
Interpreting vector equations is fundamental for determining whether two geometric entities, like lines and planes, intersect or are parallel, by evaluating their directional properties.

Intersection of Plane and Line

Finding the intersection of a line and a plane involves determining a point where the line penetrates the plane, if such a point exists. This is achieved by substituting the vector equation of the line into the equation of the plane.
Mathematically, it involves replacing the values of \( x, y, \) and \( z \) from the line's equation into the plane's equation and solving for the parameter \( t \).

• If a unique \( t \) satisfies the equation, the plane and the line intersect at a single point.
• If no such \( t \) exists, the line and the plane do not intersect.

The case where the line and the plane do not intersect occurs when the direction vector of the line is orthogonal to the normal vector of the plane, making the line parallel to the plane. Locating intersection points helps in understanding spatial relationships and is important in fields like computer graphics and geometric modeling.

Orthogonality

Orthogonality is a key concept in linear algebra that describes the relationship between two vectors that are perpendicular to each other. When two vectors are orthogonal, their dot product is zero.
In this context, the orthogonality condition \( \mathbf{v} \cdot \langle a, b, c \rangle = 0 \) indicates that the line with direction vector \( \mathbf{v} \) is perpendicular to the plane's normal vector \( \langle a, b, c \rangle \).

• This orthogonal relationship results in the line being parallel to the plane.
• No solutions exist for the parameter \( t \), meaning there are no intersection points.

Understanding orthogonality is crucial for analyzing geometric configurations and vector directions, and it is widely applied in areas such as engineering, physics, and computer science.