Problem 1 - Thomas Calculus

Question

Find parametric equations for the lines in Exercises 1–12. The line through the point \(P(3,-4,-1)\) parallel to the vector \(\mathbf{i}+\mathbf{j}+\mathbf{k}\)

Step-by-Step Solution

Answer

The parametric equations are: \(x=3+t\), \(y=-4+t\), \(z=-1+t\).

Step 1: Understanding the problem

We need to find the parametric equations of a line that passes through a given point \( P(3, -4, -1) \) and is parallel to a given vector \( \mathbf{i} + \mathbf{j} + \mathbf{k} \). In vector form, the direction vector of the line is \( \langle 1, 1, 1 \rangle \).

Step 2: Writing the direction vector

The direction vector of the line is \( \mathbf{d} = \langle 1, 1, 1 \rangle \). This means that for each unit we move along the line, \(x\), \(y\), and \(z\) increase by 1.

Step 3: Writing the parametric equations

The parametric equations of a line can be written as follows using the point \((x_0, y_0, z_0)\) and the direction vector \( \langle a, b, c \rangle \):\[ x = x_0 + at \]\[ y = y_0 + bt \]\[ z = z_0 + ct \]Substitute \(x_0 = 3\), \(y_0 = -4\), \(z_0 = -1\), and the direction vector \(\langle 1, 1, 1 \rangle\) into these formulas.

Step 4: Substitution and solving

Substitute the values into the equations:\[ x = 3 + 1t \]\[ y = -4 + 1t \]\[ z = -1 + 1t \]This gives us the parametric equations of the line.

Key Concepts

Vector Calculus

Vector calculus is a significant field in mathematics used for dealing with vectors in various applications. It helps in understanding how vectors behave, particularly when they are used to denote movement and directions in space.
Vectors are objects that have both magnitude and direction. In three-dimensional space, they are often denoted by components along the axes, such as \( \langle a, b, c \rangle \).
Instead of dealing merely with numbers, vector calculus allows one to comprehend physical phenomena like flow and force fields. This makes it incredibly useful in physics and engineering.

• For example, a vector \( \langle 1, 1, 1 \rangle \) represents a uniform direction in 3D space.
• When we say a vector is parallel to a line, it means the line's direction matches the vector's direction.

Mastering vector calculus involves learning how to manipulate vectors using operations like addition, subtraction, and scalar multiplication.

Direction Vector

The direction vector is fundamental in defining the orientation of a line in space. It tells us which path a line follows and gives a blueprint of its structure.
This vector is represented typically as \( \langle a, b, c \rangle \), where \( a, b, \) and \( c \) specify how much the line changes in the x, y, and z directions respectively.
In essence:

• If your line's direction vector is \( \langle 1, 1, 1 \rangle \), each variable (x, y, z) increases by 1 as you move along the line.
• The direction vector is crucial when determining parallelism. Two lines with the same direction vector are parallel.

The beauty of using a direction vector is it simplifies complex vector problems by highlighting the line's trajectory in space.

Line Equations

Line equations are the mathematical representation of lines in space. With vectors, one can express these lines succinctly using a point and a direction vector.
To illustrate a line passing through a point \( (x_0, y_0, z_0) \) using a direction vector \( \langle a, b, c \rangle \), we use the parametric equations:

• \( x = x_0 + at \)
• \( y = y_0 + bt \)
• \( z = z_0 + ct \)

Here, \( t \) is a parameter that adjusts how far we travel along the line.
By substituting the correct values into these equations, you can pinpoint any location on the line:

• For example, substituting point \( (3, -4, -1) \) and direction vector \( \langle 1, 1, 1 \rangle \), the equations become
  \( x = 3 + t \), \( y = -4 + t \), and \( z = -1 + t \).

These equations are essential tools for finding positions on lines in geometric and calculus contexts.

Parametric Form

The parametric form of a line converts and simplifies line equations by employing parameters like \( t \), providing clear visualization and computation.
In defining lines, the parametric form allows each coordinate (x, y, z) to be expressed as independent equations based on a parameter, typically \( t \).
This form is particularly advantageous because:

• It easily handles multiple dimensions, from simple 2D to complex 3D problems.
• It provides a versatile approach for dealing with curves and motions in geometry.

Understanding parametric form means recognizing that any point on the line results from selecting a specific \( t \) value. For the line through \( P(3, -4, -1) \) with direction \( \langle 1, 1, 1 \rangle \), varying \( t \) gives all potential points on the line.
This method is very effective in situations where the line's direction and starting point are known, and you wish to explore its entire path.