Question

The vertices of u tetrahedron are four rertices of a cube symmetrically chosen so that no two are adjacent. Show that the angle hetween the vertices of a tetrahedron is \(109.5^{\circ}\).

Short answer

The angle between the vertices of a tetrahedron, derived from a cube as described above, is approximately \(109.5^{\circ}\).\n

Step-by-step solution

Setting up the problem

First, let's visualize the problem. A cube has eight corners. If we now imagine a regular tetrahedron (a pyramid with a triangular base) inscribed in the cube, and take four non-adjacent corners of the cube, we can see that they form the vertices of the tetrahedron.

Recognizing the vectors

If we look closely, we can say that the four vectors that point from the center of the cube to the vertices of the tetrahedron are \(x\), \(y\), \(z\), \(-x-y-z\). Without loss of generality, let the vectors be unit vectors.

Calculating the dot product

Let's calculate the dot product between any two vectors. For instance, \(x\) and \(y\), we get \(x\cdot y = |x||y|\cos(\theta)\), where \(|x|\) and \(|y|\) are the magnitudes of \(x\) and \(y\) respectively and \(\theta\) is the angle between \(x\) and \(y\). Since \(x\) and \(y\) are orthogonal, their dot product will be zero.

Calculating the angle

The angle that we are interested in is formed between the vectors \(x\) and \(-x-y-z\). So, we need to calculate the dot product of \(x\) and \(-x-y-z\). Setting that equal to the magnitude of \(x\) and \(-x-y-z\) multiplied together and cosine of the angle we are looking for will yield: \(\cos(\theta) = \frac{x\cdot(-x-y-z)}{|x||-x-y-z|} = -\frac{1}{\sqrt{3}}\). Solving this for \(\theta\) gives \( \theta = \arccos(-1/\sqrt{3}) \approx 109.5 ^\circ \).

Key concepts

Cube Symmetry

Symmetry in geometry refers to a figure that is invariant under certain transformations, such as rotations or reflections. When considering a cube, we are dealing with a shape that has a high degree of symmetry. This is because a cube can be rotated in several ways — around its axes, along its edges, or about its planes — and it will still look the same.

In the context of a tetrahedron, symmetry plays a crucial role because it involves selecting vertices that exhibit specific symmetrical relationships. The importance of symmetry becomes evident when choosing the four vertices from the cube to form a tetrahedron. Here, these points must be equidistant, maintaining a symmetrical arrangement.

• This means each pair of chosen vertices is diagonally opposite.
• Achieving such symmetry is vital to ensure that the edges of the tetrahedron are equal.

These concepts of symmetry are foundational in constructions involving symmetrical shapes like cubes and tetrahedrons.

Dot Product

The dot product is a mathematical operation that takes two equal-length sequences of numbers and returns a single number. It is widely used in geometry to find the angle between two vectors. In our scenario of a tetrahedron inscribed in a cube, the dot product helps in understanding the relationship between vectors emanating from the center of the cube to the vertices of the tetrahedron.

The formula for the dot product of two vectors, \(\vec{a}\) and \(\vec{b}\), is given by: \[\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(\theta),\]where \(\theta\) is the angle between the vectors.

• Zero dot product: If the vectors are orthogonal (at a right angle), their dot product is zero, which simplifies calculations.
• Significance: The sign of the dot product can tell us about the direction of vectors (positive if they point generally in the same direction, negative if opposite).

Understanding the dot product is crucial, as it helps derive the angle between vectors forming the tetrahedron's edges.

Angle Calculation

Calculating the angle between vertices of the tetrahedron requires understanding the relationships between their connecting vectors. In geometry, specifically related to our tetrahedron problem, angles between vectors are derived using the dot product formula. The challenge lies in expressing these relationships correctly.

For instance, if vectors are represented as \(\vec{a}\) and \(\vec{b}\), and the dot product is known, the angle \(\theta\) between them can be calculated by rearranging the dot product formula:\[\cos(\theta) = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|}.\]The resulting angle when solving \(\cos^{-1}(-1/\sqrt{3})\) gives \(\theta \approx 109.5^\circ\).

Decomposing these computations:

• The precise choice of vectors reflects the symmetry exploited in the cube.
• Acknowledge interference from directionality of vectors (signs matter).

By systematically breaking down vector relationships, we can compute vital angles characterizing the geometry of the tetrahedron.

Vertex Geometry

Vertex geometry refers to the study of the spatial arrangement of points where two or more edges meet. In the tetrahedron formed within a cube, understanding vertex geometry means understanding how these points relate to the overall 3D structure.

Vertices in this context is crucial to fully grasp the tetrahedron's formation, which involves four vertices of the cube, none of which are adjacent. This selection ensures the tetrahedron forms equilateral triangles at its faces.

• An equilateral face means each angle is \(60^\circ\), resulting from its three identical sides.
• The placement away from adjacent vertices forms equal edge lengths in the 3D shape.

Furthermore, by manipulating these vertices and transforming the cube's inherent symmetry, we gain insights into creating intricate 3D shapes.