Problem 32 - Precalculus Mathematics for Calculus

Question

A description of a line is given. Find parametric equations for the line. The plane that crosses the \(x\) -axis where \(x=-2,\) the \(y\) -axis where \(y=-1,\) and the \(z\) -axis where \(z=3\)

Step-by-Step Solution

Answer

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

Step 1: Find the points where the plane intersects the axes

The plane crosses the x-axis at the point \((-2, 0, 0)\), it crosses the y-axis at \((0, -1, 0)\), and it crosses the z-axis at \((0, 0, 3)\).

Step 2: Find the direction vector of the line

Using two of the points found, select \((-2, 0, 0)\) and \((0, -1, 0)\) and find the direction vector of the line by subtracting the coordinates: \((0 - (-2), -1 - 0, 0 - 0) = (2, -1, 0)\).

Step 3: Find the parametric equations of the line

Using one point, such as \((-2, 0, 0)\), and the direction vector \((2, -1, 0)\), the parametric equations of the line can be formed:\[x = -2 + 2t\y = 0 - t\z = 0\]Note that the z-component remains constant at zero, as it does not change along the line.

Key Concepts

Direction Vector

A direction vector plays a critical role in understanding parametric equations, which describe lines in three-dimensional space. In simple terms, a direction vector indicates the orientation of a line by showing the extent and direction in which the line extends.
To find a direction vector for a line, you subtract the coordinates of two points lying on the line. For example, given two points, \((-2, 0, 0)\) and \((0, -1, 0)\), we derive the direction vector by a component-wise subtraction:

• Subtract the x-coordinates: \(0 - (-2) = 2\)
• Subtract the y-coordinates: \(-1 - 0 = -1\)
• Subtraction of z-coordinates gives \(0 - 0 = 0\)

This results in the direction vector \((2, -1, 0)\).
This vector not only illustrates the line's path but also reflects how the line progresses in space.
By understanding this concept, you'll grasp how altering direction vectors can modify a line's path in coordinate geometry.

Intersection Points

Intersection points are the specific locations where a geometric entity, like a line or a plane, crosses the axes in a coordinate system. These points are essential because they provide positional information that helps define the entity in space.
In the context of the given problem, the plane crosses:

• The x-axis at point \((-2, 0, 0)\)
• The y-axis at point \((0, -1, 0)\)
• The z-axis at point \((0, 0, 3)\)

These intersection points are vital as they offer a concrete reference for constructing lines or further geometric components.
By using these intersection points, we can effectively determine how space and its dimensions are articulated within our mathematical models.

Coordinate Geometry

Coordinate geometry, an essential branch of mathematics, explores the relationships between geometric figures and their positions on the coordinate plane. It is instrumental in analyzing shapes, sizes, and other spatial properties using algebraic equations.
Through coordinate geometry, we can graphically represent geometric entities and evaluate their interactions. For example, the parametric equations derived from the line passing through points on the plane offer an understanding of its path.

• \(x = -2 + 2t\) indicates changes along the x-axis
• \(y = 0 - t\) describes movement along the y-axis
• \(z = 0\) signifies no change along the z-axis, meaning the line lies flat in the xy-plane

Such equations provide mathematics with a powerful toolkit to model and resolve complex problems involving dimensional interactions.
By manipulating and solving these equations, one can unlock deeper insights into the nature of space and movement within our three-dimensional world.