Problem 8 - University Calculus: Early Transcendentals

Question

Find parametric equations for the lines. The line through (2,4,5) perpendicular to the plane \(3 x+7 y\) \(-5 z=21\)

Step-by-Step Solution

Answer

The parametric equations are: \(x = 2 + 3t\), \(y = 4 + 7t\), \(z = 5 - 5t\).

Step 1: Find the normal vector of the plane

The normal vector of the plane is given by the coefficients of the variables in the plane equation. For the plane equation \(3x + 7y - 5z = 21\), the normal vector is \(\langle 3, 7, -5 \rangle\). This vector is perpendicular to the plane and will be the direction vector for our line.

Step 2: Write the parametric equations using the point and direction vector

A line through a point \((x_0, y_0, z_0)\) with direction vector \(\langle a, b, c \rangle\) can be expressed as parametric equations:\[ x = x_0 + at \]\[ y = y_0 + bt \]\[ z = z_0 + ct \]Substitute \((2, 4, 5)\) as your point and \(\langle 3, 7, -5 \rangle\) as your direction vector into the parametric form:\[ x = 2 + 3t \]\[ y = 4 + 7t \]\[ z = 5 - 5t \]These parametric equations represent the line through the given point and perpendicular to the plane.

Key Concepts

Vector Calculus

Vector calculus is a branch of mathematics focused on vector fields and operations applied to them. It's like a toolbox that helps in dealing with vectors in multidimensional spaces.

• Vectors are quantities with both magnitude and direction.
• Vector calculus includes operations such as dot product, cross product, and differentiation and integration of vectors.

In this context, we're looking at a line and plane in three-dimensional space. Vector calculus lets us analyze such relationships effectively. Understanding how vectors describe lines and planes helps us solve problems like finding parametric equations for lines. It forms the groundwork to be able to manipulate and calculate various properties such as perpendicularity and direction of vectors.

Direction Vector

A direction vector is a fundamental concept when describing lines. It provides the direction in which the line extends in space. For example, if the direction vector is \( \langle a, b, c \rangle \), this means:

• The line moves \( a \) units in the x-direction.
• It moves \( b \) units in the y-direction.
• And \( c \) units in the z-direction, for each unit increment of the parameter.

In our exercise, the line is perpendicular to the given plane, and the direction vector of the line is exactly the normal vector of the plane. Thus, if the line through point \((2, 4, 5)\) takes the normal vector \( \langle 3, 7, -5 \rangle \), it moves based on these components in the 3D space.

Normal Vector

The normal vector of a plane is a vector that is perpendicular to every line in the plane. Considering the plane equation \( 3x + 7y - 5z = 21 \), the normal vector is \( \langle 3, 7, -5 \rangle \). This vector is not just important for the plane but it also dictates that any line using this vector as a direction vector will be perpendicular to the plane.

• It is obtained directly from the coefficients of \( x, y, \) and \( z \) in the plane equation.
• The normal vector tells us how the plane is oriented in space.

This concept helps identify the perpendicular relationship between the plane and the line, meaning they intersect at 90-degrees, or are orthogonal.

Perpendicular Line

The concept of a perpendicular line involves two lines intersecting at a right angle. In our exercise, the line sought is perpendicular to the plane. This is achieved with the direction vector of the line being the normal vector of the plane.

• A line is perpendicular to a plane if its direction vector aligns with the plane's normal vector.
• This ensures the line and plane meet at 90 degrees, a fundamental geometric property.

Thus, the parametric equations derived, such as \( x = 2 + 3t \), \( y = 4 + 7t \), and \( z = 5 - 5t \), represent a line perfectly aligned and perpendicular to the given plane. Ensuring the conceptual understanding of perpendicularity gives insight into spatial relationships in vector geometry.