Question

Find the volume of the four-dimensional pyramid bounded by \(w+x+y+z+1=0\) and the coordinate planes \(w=0, x=0, y=0,\) and \(z=0\).

Short answer

Answer: The volume of the four-dimensional pyramid is 2 cubic units.

Step-by-step solution

Determine the vertices of the pyramid

To find the intersection of the hyperplane with each of the four coordinate planes, we will substitute the respective variables with zero and solve for the remaining variable: 1. Intersection with \(w=0\): \((0, x, y, z) \Rightarrow x + y + z = -1\) 2. Intersection with \(x=0\): \((w, 0, y, z) \Rightarrow w + y + z = -1\) 3. Intersection with \(y=0\): \((w, x, 0, z) \Rightarrow w + x + z = -1\) 4. Intersection with \(z=0\): \((w, x, y, 0) \Rightarrow w + x + y = -1\) Now we solve each system of equations: 1. \((0, -1, 0, 0)\) 2. \((-1, 0, 0, 0)\) 3. \((0, 0, -1, 0)\) 4. \((0, 0, 0, -1)\) So, the vertices of the four-dimensional pyramid are \((0, -1, 0, 0), (-1, 0, 0, 0), (0, 0, -1, 0), (0, 0, 0, -1)\), and \((0,0,0,0)\).

Find the vectors connecting the vertices

To find the vectors connecting the vertices, we subtract the coordinates of the origin from the coordinates of each vertex in the shape: 1. \((-1, 1, 0, 0)\) 2. \((0, 1, -1, 0)\) 3. \((0, 0, 1, -1)\) 4. \((0, 0, 0, 1)\) The vectors connecting the vertices of the four-dimensional pyramid are \((-1, 1, 0, 0), (0, 1, -1, 0), (0, 0, 1, -1)\), and \((0, 0, 0, 1)\).

Compute the volume using the determinant of the matrix

We will now calculate the volume of the pyramid by computing the determinant of the matrix formed by the vectors. The matrix is: [[-1, 1, 0, 0], [ 0, 1,-1, 0], [ 0, 0, 1,-1], [ 0, 0, 0, 1]] To compute the determinant, we can expand by co-factors along the last row: \(\text{det}(M) = 1*(\text{det}(3x3 \, \text{matrix}))\) The resulting 3x3 matrix is: [[-1, 1, 0], [ 0, 1,-1], [ 0, 0, 1]] Calculating the determinant of the 3x3 matrix, we obtain: \(\text{det}(3x3) = -1*(-1 - (-1)) = -1*(-2) = 2\) Thus, the working of the determinant of the original 4x4 matrix is: \(\text{det}(M) = 1*(2) = 2\) The volume of the four-dimensional pyramid is 2 cubic units.

Key concepts

Volume of a Hyperpyramid

In four-dimensional space, a hyperpyramid is similar to a three-dimensional pyramid, but it exists in an extra dimension. Determining the volume of such a shape requires understanding the relationships and intersections between planes and coordinate systems in this expanded space. The hyperpyramid's volume relates to the determinant of a matrix formed by vectors connecting the pyramid's origin to its vertices.

The volume of a hyperpyramid can be determined by the formula:

• Volume = \( \frac{1}{n!} \times \text{det}(M) \)
• In this case, \( n = 4 \), representing four dimensions.

For our example, the determinant of the \(4 \times 4\) matrix is 2, so the volume is:

• Volume = \( \frac{1}{4!} \times 2 = \frac{1}{24} \times 2 = \frac{1}{12} \) cubic units.

This calculation illustrates the unique complexities of working with dimensions beyond those we can visualize easily.

Coordinate Planes

Coordinate planes in four-dimensional space serve as boundaries for hyperplanes, dividing this space into regions. Each coordinate plane is defined by one variable being equal to zero. For example, the plane \(w=0\) in 4D coordinates resets the first variable to zero while allowing \(x, y,\) and \(z\) to vary. These planes help define the vertices of geometric objects like hyperpyramids.

In our exercise, the coordinate planes intersect with the hyperplane defined by \(w + x + y + z + 1 = 0\). By setting one variable to zero at a time, we find intersections that help locate the pyramid's vertices:

• For \(w=0\), the resulting plane equation is \(x + y + z = -1\).
• For \(x=0\), it becomes \(w + y + z = -1\).
• For \(y=0\), it turns into \(w + x + z = -1\).
• For \(z=0\), equation is \(w + x + y = -1\).

These intersections create a tangible sense of structure within the abstract space, allowing us to visualize and solve geometric problems.

Determinants

Determinants are a powerful mathematical tool used in linear algebra to understand the properties of matrices. In our context, they help calculate volumes of multi-dimensional shapes. A determinant of a matrix gives insight into the scale factor for the transformation represented by the matrix.

To find the volume of our hyperpyramid, we looked at the determinant of a matrix comprising vectors from the pyramid's origin to its vertices. This process involved creating a \(4 \times 4\) matrix from these vectors and calculating its determinant, which simplifies to finding a \(3 \times 3\) matrix determinant.

• The determinant of the \(3 \times 3\) matrix is computed as \(-1(-1 - (-1)) = 2\).
• This result helps establish the volume of our four-dimensional structure.

Determinants essentially condense complex spatial relationships into a single value, illustrating how these multidimensional shapes behave or transform.

Vectors in Higher Dimensions

Vectors are lines of magnitude and direction, and when extended into higher dimensions, they offer a method to navigate and calculate within complex spaces. In four-dimensional geometry, vectors connect two points in this extended space, making them crucial for defining shapes like hyperpyramids.

The vectors in our problem result from subtracting the origin point's coordinates from the pyramid's vertices' coordinates. These vectors are:

• From the origin to \((-1, 1, 0, 0)\)
• To \((0, 1, -1, 0)\)
• To \((0, 0, 1, -1)\)
• To \((0, 0, 0, 1)\)

These vectors form the basis for building the matrix whose determinant we calculate to find the volume. Understanding vectors in this context allows one to see how points, lines, and shapes exist within a space that goes beyond our mundane three-dimensional experiences. This knowledge is crucial for linking algebraic manipulations to geometric transformations and problem-solving.