Problem 26 - Algebra and Trigonometry

Question

Find parametric equations for the line with the given properties. Passing through \((12,7)\) and the origin

Step-by-Step Solution

Answer

The parametric equations are \( x = 12t \) and \( y = 7t \).

Step 1: Define the Points

The line passes through the points \((12,7)\) and \((0,0)\). These points provide us with positional information needed to define the line parametrically.

Step 2: Determine the Direction Vector

To find the direction vector \( \mathbf{d} \) of the line, we calculate the difference between the coordinates of the two points. The direction vector \( \mathbf{d} \) will be \( (12 - 0, 7 - 0) \), which simplifies to \( \mathbf{d} = (12, 7) \).

Step 3: Write the Parametric Equations

Let \( t \) be the parameter. A point on the line can be represented using the vector equation: \[\mathbf{r}(t) = \mathbf{r}_0 + t \cdot \mathbf{d}\]where \( \mathbf{r}_0 \) is the position vector of the starting point, here \((0,0)\), and \( \mathbf{d} = (12, 7) \). Therefore, the parametric equations are:\[ x = 0 + 12t = 12t \]\[ y = 0 + 7t = 7t \]

Key Concepts

Direction Vector

In the world of vectors and lines, a direction vector is vital. It tells us the 'direction' in which the line moves. For any given line, the direction vector is derived by subtracting the coordinates of two points on the line.
In simpler terms, it's like pointing your finger along the path of the line. Imagine a line passing through point A(12, 7) and point B(0, 0). To get the direction vector, think of moving from B to A. By subtracting the coordinates:

• From the x-values: \(12 - 0 = 12\)
• From the y-values: \(7 - 0 = 7\)

This gives us the direction vector \((12, 7)\).
It's important because all moves along this line will align with this vector. It helps in defining where the line "goes" beyond the given points.

Vector Equation

The vector equation of a line neatly ties together the direction vector and a point on the line. This equation serves as the backbone for creating parametric equations that define the line in mathematical terms.The vector equation can be written as: \[\mathbf{r}(t) = \mathbf{r}_0 + t \cdot \mathbf{d} \]Here:

• \( \mathbf{r}(t) \) is the general position vector of any point on the line.
• \( \mathbf{r}_0 \) is the position vector of a known point on the line, typically the starting point or origin.
• \( t \) is the parameter, a variable that influences how far along the direction vector the point is.
• \( \mathbf{d} \) is the direction vector.

By substituting \( \mathbf{r}_0 = (0, 0) \) and \( \mathbf{d} = (12, 7) \), the vector equation becomes practical and allows computation of any point on the line as the parameter \( t \) varies.

Position Vector

The concept of a position vector is foundational when dealing with vector equations and parametric forms. It represents a fixed point in space, defined in terms of coordinates.For a line, choosing a position vector is like picking a starting point from which all other points on the line will be plotted. In our example, this starting point is the origin, noted as \( (0, 0) \). From this origin, the position vector is simply \( \mathbf{r}_0 = (0, 0) \).
The role of the position vector is crucial in parametric equations. It sets the base location on the line and, with the addition of a scaled direction vector, enables finding any point on the line. For instance:

• The x-parametric equation is derived as \( x = 0 + 12t \)
• The y-parametric equation comes from \( y = 0 + 7t \)

By starting from the origin, indicated by the position vector, these parametric equations describe how you move along the line based on different values of \( t \).