Problem 23 - Algebra and Trigonometry

Question

Visualizing a Set in Space Try to visualize the set of all points \((x, y, z)\) in a coordinate space that are equidistant from the points \(P(0,0,0)\) and \(Q(0,3,0) .\) Use the Distance Formula to find an equation for this surface, and observe that it is a plane.

Step-by-Step Solution

Answer

The equidistant points form the plane \(y = \frac{3}{2}\).

Step 1: Understanding the problem

We need to find the set of all points in 3D space that are equidistant from two given points, \(P(0,0,0)\) and \(Q(0,3,0)\). This means any point \((x, y, z)\) on this set satisfies \(d(P,(x, y, z)) = d(Q,(x, y, z))\).

Step 2: Applying the Distance Formula

The distance from a point \((x, y, z)\) to \(P(0,0,0)\) is given by the formula \(\sqrt{x^2 + y^2 + z^2}\). Similarly, the distance to \(Q(0,3,0)\) is \(\sqrt{x^2 + (y-3)^2 + z^2}\).

Step 3: Setting the Distances Equal

Since the points are equidistant from \(P\) and \(Q\), we equate the two distance expressions: \[ \sqrt{x^2 + y^2 + z^2} = \sqrt{x^2 + (y-3)^2 + z^2} \]

Step 4: Eliminating the Square Roots

Square both sides of the equation to eliminate the square roots: \[ x^2 + y^2 + z^2 = x^2 + (y-3)^2 + z^2 \]

Step 5: Simplifying the Equation

Subtract \(x^2\), \(z^2\) from both sides to simplify: \[ y^2 = (y-3)^2 \]. Expand \((y-3)^2\): \[ y^2 = y^2 - 6y + 9 \].

Step 6: Solving for y

Cancel \(y^2\) from both sides, resulting in \[ 0 = -6y + 9 \]Rearrange to solve for \(y\):\[ 6y = 9 \]\[ y = \frac{3}{2} \].

Step 7: Concluding the Visualization

Thus, the set of all points equidistant from \((0,0,0)\) and \((0,3,0)\) is the plane \(y = \frac{3}{2}\), parallel to the xz-plane.

Key Concepts

Distance Formula

The distance formula is a key concept in coordinate geometry, used to calculate the distance between two points in space. In 3D space, if we have two points \(A(x_1, y_1, z_1)\) and \(B(x_2, y_2, z_2)\), the distance \(d\) between them is given by:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \] This formula is an extension of the Pythagorean theorem and applies to three perpendicular axes for spatial calculations.
When solving problems involving equidistant points, like in our exercise, using the distance formula helps us set up equations that show when two distances are equal. Learning how to apply and manipulate the formula is essential for solving geometric problems in 3D.

3D Coordinate Space

3D coordinate space adds another dimension to the familiar 2D plane, giving us a richer way to represent points. Every point in this space is specified by three coordinates \( (x, y, z) \), each representing a position along the Cartesian axes. These axes are typically referred to as the x-axis, y-axis, and z-axis.
In 3D space:

• The x-axis typically extends left and right.
• The y-axis extends up and down.
• The z-axis extends toward and away from you.

Understanding how to navigate this space is crucial for visualizing geometric shapes and solving equations describing various surfaces and lines. This further enhances our problem-solving capabilities.

Geometric Visualization

Geometric visualization in 3D enables you to intuitively understand complex spatial relationships. By conceptualizing coordinate points as actual locations within a 3D space, one can better appreciate the physical aspects of geometry.
In our particular exercise, geomteric visualization helps us understand that a set of equidistant points forms a plane. This plane is parallel to one of the coordinate axes.

• It emphasizes the locus of points sharing equal distance properties.
• Visualization aids in recognizing patterns and symmetry in geometry, pivotal in solving spatial problems.

Developing the skill to visualize in 3D is integral not just for geometry but also fields like physics, engineering, and computer graphics.

Equidistant Points

Equidistant points refer to a set of points that share the same distance from two or more locations. The concept forms the basis for understanding how geometric places like planes or circles are derived from specific conditions.
In the context of our exercise, finding the set of points equidistant from \(P(0,0,0)\) and \(Q(0,3,0)\) results in a plane in space. To find this plane, we set up an equation where distances from these points are equal, leading to the equation \(y = \frac{3}{2}\).

• This equation describes an infinite set of points, forming a flat surface, or a plane.
• This concept is used extensively in creating mirrors, bisectors, and understanding symmetry.

Mastering this concept allows for better comprehension of spatial equality and symmetry in geometric problems.