Question

The vertices of a tetrahedron correspond to four alternating corners of a cube. By using analytical geometry, demonstrate that the angle made by connecting two of the vertices to a point at the center of the cube is \(109.5^{\circ}\), the characteristic angle for tetrahedral molecules.

Short answer

The angle made by connecting two alternating vertices of a cube to the center point can be found using analytical geometry. By assigning coordinates to the vertices of the cube and choosing the alternating vertices, we can find the center point of the cube and the position vectors from the center to the chosen vertices. After calculating the dot product and the magnitudes of the position vectors, we use the dot product formula to find the angle between them, which is approximately \(109.5^{\circ}\). This is the characteristic angle for tetrahedral molecules.

Step-by-step solution

Assign coordinates to the vertices of the cube

Let's assign coordinates to the vertices of the cube, starting at one corner and taking the cube's side as length 1. The vertices of the cube can be represented as: A (0, 0, 0), B (1, 0, 0), C (1, 1, 0), D (0, 1, 0), E (0, 0, 1), F (1, 0, 1), G (1, 1, 1), and H (0, 1, 1)

Choose the alternating vertices

Now we need to choose four alternating vertices to create a tetragon. Let's select vertices A, C, E, and G. This way we have: A (0, 0, 0), C (1, 1, 0), E (0, 0, 1), and G (1, 1, 1)

Find the center of the cube

The center of the cube can be found by taking the average of the coordinates of opposing vertices. The coordinates of the center point O are: O \(\left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)\)

Find the position vectors

We need the position vectors from the center of the cube (O) to the vertices A and G. Using the coordinates of O, A, and G, we find: \(\vec{OA} = \begin{bmatrix}-\frac{1}{2} \\ -\frac{1}{2} \\ -\frac{1}{2} \end{bmatrix}\) and \(\vec{OG} = \begin{bmatrix}\frac{1}{2} \\ \frac{1}{2} \\ \frac{1}{2} \end{bmatrix}\)

Calculate the dot product

Now, we can find the dot product of the two position vectors: \(\vec{OA} \cdot \vec{OG} = -\frac{1}{2}\times\frac{1}{2} + (-\frac{1}{2})\times\frac{1}{2} + (-\frac{1}{2})\times\frac{1}{2} = -\frac{1}{4}\)

Calculate the magnitudes of the position vectors

Next, we need to find the magnitudes of the position vectors \(\vec{OA}\) and \(\vec{OG}\): \(|\vec{OA}| = |\vec{OG}| = \sqrt{\left(-\frac{1}{2}\right)^2 + \left(-\frac{1}{2}\right)^2 + \left(-\frac{1}{2}\right)^2} = \frac{\sqrt{3}}{2}\)

Use the dot product formula to find the angle

Now, we can use the dot product formula to find the angle between the two position vectors. The dot product formula is: \(\cos \theta = \frac{\vec{OA} \cdot \vec{OG}}{|\vec{OA}||\vec{OG}|}\) Substitute the known values into the formula: \(\cos \theta = \frac{-\frac{1}{4}}{\frac{\sqrt{3}}{2} \times \frac{\sqrt{3}}{2}} = -\frac{1}{3}\) Now find the angle \(\theta\) by taking the inverse cosine: \(\theta = \cos^{-1} (-\frac{1}{3})\) \(\theta \approx 109.5^{\circ}\) The angle made by connecting two of the vertices to the center of the cube is approximately \(109.5^{\circ}\), which is the characteristic angle for tetrahedral molecules.

Key concepts

Tetrahedron

A tetrahedron is a type of polyhedron composed of four triangular faces. It's the simplest 3-dimensional shape in geometry. A key feature of a tetrahedron is that it has four vertices and six edges, with three edges converging at each vertex. Each face is a triangle, making the tetrahedron a type of triangular pyramid.

In an exercise involving cube geometry as described, tetrahedrons are formed by picking four vertices of a cube that are not on the same face. In this scenario, the chosen vertices form an alternating pattern within the cube. Each face of the tetrahedron is equilateral if all cube edges are equal. Consequently, all internal angles between its planes are equal as well, leading to the characteristic tetrahedral angle of approximately 109.5°.

Understanding tetrahedrons involves knowing the properties of all its constituting elements like faces, vertices, and edges. Tetrahedrons as a concept also extend into molecular geometry, describing structures where four atoms are equally spaced around a central atom.

Tetrahedral Angle

In geometry, a tetrahedral angle is the angle between any two lines drawn from a vertex to the centroid of a tetrahedron, meeting at equal angles with adjacent lines. This angle is classically uniform across a perfect tetrahedron.

For example, within a cube, if vertices are selected in an alternating pattern to create a tetrahedron, the tetrahedral angle measures approximately 109.5°. This angle becomes significant, as it is also found in chemical bonding of molecules like methane (CH₄).

The tetrahedral angle is calculated using analytical geometry by considering the position vectors from a central point to the vertices of the tetrahedron. The angle is derived mathematically using vector properties and trigonometry as demonstrated within the exercise, particularly through the use of the dot product and inverse cosine function.

Cube Geometry

Cube geometry is vital in understanding many 3-dimensional configurations. A cube is a regular polyhedron with six identical square faces, twelve equal edges, and eight vertices. It is defined in space by three perpendicular axes.

When considering cube geometry relative to tetrahedrons, the focus is often on the spatial relationships and coordinates of its vertices. Analyzing the cube's vertices allows us to derive properties of inscribed shapes, such as tetrahedrons. For example, in this exercise, analyzing the vertices helps identify the center of the cube, crucial for determining the tetrahedral angle.

Often, problems in cube geometry are solved by assigning a coordinate system, which facilitates the calculation of distances and angles through linear algebra and vector analysis.

Position Vectors

Position vectors are an integral part of analytical geometry as they provide a way to describe the position and movement within a coordinate system. A position vector originates from a fixed point, typically designated as the origin, and points towards a specific position in space.

In geometric problems like the one given in the exercise, position vectors are used to connect different points in the 3D space of a cube. For instance, the vector from the center of the cube to a vertex helps determine the angle between lines to vertices, essentially leading to the calculation of the tetrahedral angle.

Using position vectors, one can compute vector magnitude and direction, find angles between vectors using dot products, and effectively solve spatial problems. They play a crucial role in bridging the gap between the geometry of a shape and its algebraic representation.