Problem 6 - Algebra and Trigonometry

Question

Find parametric equations for the line that passes through the point \(P\) and is parallel to the vector \(\mathbf{v} .\) $$ P(0,0,0), \quad \mathbf{v}=\langle- 4,3,5\rangle $$

Step-by-Step Solution

Answer

The parametric equations are \(x = -4t\), \(y = 3t\), and \(z = 5t\).

Step 1: Identify the Components

To find the parametric equations for a line, we need two things: a point on the line and a direction vector. From the problem, the point is given as \(P(0,0,0)\) and the direction vector \(\mathbf{v}=\langle -4, 3, 5 \rangle\).

Step 2: Write the Parametric Equations

The parametric equations of a line in three-dimensional space can be written as \( x = x_0 + at \), \( y = y_0 + bt \), \( z = z_0 + ct \), where \((x_0, y_0, z_0)\) is a point on the line, and \(\langle a, b, c \rangle\) is the direction vector. In this case, we have \(x_0 = 0\), \(y_0 = 0\), \(z_0 = 0\), and \(\langle a, b, c \rangle = \langle -4, 3, 5 \rangle\).

Step 3: Substitute the Values

Substitute the values from Step 2 into the parametric equations. Thus, the equations become \( x = 0 - 4t \), \( y = 0 + 3t \), and \( z = 0 + 5t \).

Step 4: Simplify the Equations

Simplifying the equations from Step 3, we get \( x = -4t \), \( y = 3t \), and \( z = 5t \) as the parametric equations for the line.

Key Concepts

Direction Vector

In geometry, the direction vector plays a crucial role in defining the orientation of a line in space. It gives the path the line follows by showing the line's direction along each axis.
For a line in three-dimensional space, the direction vector is represented by \(\mathbf{v} = \langle a, b, c \rangle\). Here, \(a\), \(b\), and \(c\) are the components that denote how much the line moves in the \(x\)-, \(y\)-, and \(z\)-directions, respectively.

• \(a\) indicates horizontal movement along the \(x\)-axis
• \(b\) indicates vertical movement along the \(y\)-axis
• \(c\) indicates depth or out-of-plane movement along the \(z\)-axis

By multiplying these components by a variable \(t\), which is a parameter, the line can represent every point along its infinite length. Therefore, the direction vector not only tells us how the line tilts or shifts in three-dimensional space but also helps to create the parametric equations of the line itself.

Three-dimensional Space

Three-dimensional (3D) space is the environment in which we live and interact daily, and it consists of three axes: \(x\), \(y\), and \(z\). These axes provide coordinates which help to locate points, lines, and shapes in space.
When defining a line in this space, we need at least one point on the line and a direction vector, allowing us to express the line using parametric equations.
These equations take the form:

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

In these equations, \((x_0, y_0, z_0)\) is the initial point on the line, while \(\langle a, b, c \rangle\) is the direction vector. By varying \(t\), the position changes, representing all possible points on the line through this infinite space. Understanding how lines and other geometric objects are oriented in three-dimensional space is essential for applications ranging from engineering designs to video game graphics.

Point on the Line

To fully describe a line in space using parametric equations, you first need a specific point on that line. This point is often called the anchor or reference point, because it acts as a starting place from which the line extends in the direction determined by the direction vector.
Given a point \(P(x_0, y_0, z_0)\), we can use the coordinates of this point in our parametric equations. For the exercise at hand, our point is \((0, 0, 0)\). The initial state of the line can thus be expressed as the additive term in each parametric equation since it represents where the line exactly "sits" before \(t\) starts changing:

• \(x_0 = 0\)
• \(y_0 = 0\)
• \(z_0 = 0\)

Once this point is established, the position along the line at any moment is calculated with the help of the direction vector as \(t\) varies. This combination of initial point and direction vector ensures that every possible point on the line is accessible and can be calculated precisely.