Problem 6 - APEX Calculus

Question

Give any two points in the given plane. \(4(y+2)-(z-6)=0\)

Step-by-Step Solution

Answer

Two points on the plane are \((0, 0, 14)\) and \((0, 1, 18)\).

Step 1: Understand the Plane Equation

The equation of the plane is given as \(4(y+2) - (z-6) = 0\). First, simplify this equation. Distribute the 4 to get \(4y + 8 - z + 6 = 0\). Simplify further to \(4y - z + 14 = 0\). This is the standard form of a plane equation \(Ax + By + Cz = D\), where \(A = 0\), \(B = 4\), \(C = -1\), and \(D = -14\).

Step 2: Find the First Point

To find a point on this plane, choose any two variables to set as specific values and solve for the third. Assume \(y = 0\) and \(z = 0\). Substitute these into the plane equation: \(4(0) - 0 + 14 = 0\). This simplifies to \(14 = 0\), which is not possible; therefore, adjust the assumption. Now, let \(y = 0\) and solve for \(z\): \(4(0) - z + 14 = 0\). Thus, \(z = 14\). The point is \((x, y, z) = (0, 0, 14)\).

Step 3: Find the Second Point

Choose another set of \(y\) and \(z\) values. This time, set \(y = 1\). Substitute into the plane equation: \(4(1) - z + 14 = 0\). This simplifies to \(4 - z + 14 = 0\) or \(z = 18\). Thus, the second point is \((x, y, z) = (0, 1, 18)\).

Step 4: Verify the Points

Ensure that both points satisfy the plane equation. - For \((0, 0, 14)\): Substituting into \(4y - z + 14 = 0\), we get \(4(0) - 14 + 14 = 0\), which is true.- For \((0, 1, 18)\): Substituting into the equation, \(4(1) - 18 + 14 = 0\), which is also true. Both points lie on the plane.

Key Concepts

Coordinate Geometry

Coordinate Geometry is a branch of mathematics that uses an ordered pair of numbers to represent points on a plane. It provides a connection between algebra and geometry, allowing you to solve geometrical problems using coordinate planes.
In coordinate geometry, any point in a 2D or 3D space can be represented using coordinates. For example, a point in a 2D plane has coordinates \(x, y\), while a point in a 3D space is represented by \(x, y, z\). These coordinates allow us to uniquely identify each point and explore relationships between them with geometric equations.
In the case of a plane, it extends this to three-dimensional space and helps define the relative positions and distances of points in that space. By understanding the role of coordinates, we can derive various forms of geometric equations, such as lines, planes, and more.

3D Space

3D space, or three-dimensional space, introduces a new way of looking at objects and positions beyond the conventional 2D plane. In 3D space, every point is identified by three coordinates: \(x, y, z\). This allows for more complex geometric shapes and relationships to be studied.

• \(x\) represents the horizontal distance from an origin point.
• \(y\) represents the vertical distance.
• \(z\) adds the depth dimension, making it possible to describe objects in volume.

Planes in 3D space are flat surfaces extending infinitely along the x, y, and z axes. They can be visualized as sheets of paper stretching endlessly in space, without thickness. Understanding 3D spaces is crucial because it models the real world we live in, helping us solve problems involving volume, center of mass, and more.

Equation of a Plane

The equation of a plane is a mathematical expression defining a flat surface in 3D space. The general form of a plane equation is \(Ax + By + Cz = D\), where \(A, B,\) and \(C\) are constants indicating the plane's orientation, while \D\ relates to its position in the space.
In the given problem, the plane equation is simplified to \4y - z + 14 = 0\. Here, \(A = 0\), \(B = 4\), \(C = -1\), and \(D = -14\). This indicates the plane is parallel in certain directions while having a specific tilt due to the coefficients of \(B\) and \(C\).

• The term \(4y\) implies that changes in the \(y\) direction affect the plane's characteristics.
• The \(-z\) term indicates how the \(z\) direction modifies the plane's positioning.
• The constant \(14\) can be seen as an offset, moving the plane within 3D space.

Understanding plane equations allows us to pinpoint exact coordinates of points on the plane, and determine how various parameters define the plane's tilt and position in space.