Problem 5 - University Calculus: Early Transcendentals

Question

Find parametric equations for the lines. The line through the origin parallel to the vector \(2 \mathbf{j}+\mathbf{k}\)

Step-by-Step Solution

Answer

The parametric equations are \(x = 0\), \(y = 2t\), \(z = t\).

Step 1: Understand the Line's Direction

The line goes through the origin, so the direction is determined by the parallel vector. A line parallel to vector \(\mathbf{v} = 2\mathbf{j} + \mathbf{k}\) means the direction numbers for the line are \(0\), \(2\), and \(1\) since there is no \(i\) component.

Step 2: Identify the Point on the Line

The line goes through the origin, which is the point \((0, 0, 0)\). We will use this point as the starting point in our parametric equations.

Step 3: Write the Parametric Equations of the Line

Using the direction vector \(\mathbf{v} = 0\mathbf{i} + 2\mathbf{j} + 1\mathbf{k}\) and the point \((0, 0, 0)\), the parametric equations are: \[ x = 0 + 0t = 0 \]\[ y = 0 + 2t = 2t \]\[ z = 0 + 1t = t \]

Key Concepts

Vector Algebra

Vector algebra is a branch of mathematics that deals with vectors and the rules for manipulating them. Vectors are quantities that have both magnitude and direction.
In vector algebra, vectors are often represented in the form of components along specific directions like the coordinate axes.

• A vector can be denoted by letters like \( \mathbf{i}\) for the x-axis, \( \mathbf{j}\) for the y-axis, and \( \mathbf{k}\) for the z-axis.
• Vector operations include addition, subtraction, and scalar multiplication. These operations help us perform mathematical calculations involving vectors efficiently.

Vector algebra is crucial in understanding geometrical problems that involve angles, distances, and directions. It is especially useful in representing lines and planes in three-dimensional space.

Direction Vector

The direction vector is a fundamental concept when dealing with lines in space. It designates the specific direction in which a line extends. This vector is parallel to the line and essential in forming parametric equations.
For example, in our exercise, the direction vector is \(\mathbf{v} = 2\mathbf{j} + \mathbf{k}\).

• This indicates that the line moves 2 units in the y-direction and 1 unit in the z-direction with no movement in the x-direction.
• The coefficients \(2\) and \(1\) come directly from the components of the vector that influences the parametric equations.

Understanding the direction vector is key to knowing how the line behaves in space, giving us insight into how to model it mathematically.

Origin

In mathematics, the origin is a special point that is often used as a reference when dealing with coordinate systems. It is the point where all coordinate axes intersect and is usually denoted as \((0, 0, 0)\) in three-dimensional space.
In our exercise, the origin is the point through which our line passes.

• This simplifies the process of creating parametric equations because the initial coordinates can be set as zero.
• The origin plays a significant role in various mathematical contexts, serving both as a starting point and a datum for measuring coordinates.

Understanding the origin allows for intuitive grounding in spatial reasoning and helps clarify the point of departure for lines in vector calculations.

Parametric Representation

Parametric representation allows us to describe geometric objects like lines through equations that depend on a parameter. For lines, this usually involves expressing the x, y, and z coordinates as functions of a single parameter, \(t\).
In the solution, our line's parametric equations are:

• \(x = 0\)
• \(y = 2t\)
• \(z = t\)

Each equation describes the position of the line at any point along its path by varying \(t\). This representation is especially useful because:

• It captures how a line changes its position in space over time.
• It allows for easy calculation of specific points on the line by substituting different values for \(t\).

The parametric representation is a powerful tool in computational geometry and helps create versatile models of geometric entities.