Question

Consider a rectangular solid with sides of lengths \(a, b,\) and \(c .\) Show that it has two orthogonal diagonals if and only if the sum of two of \(a^{2}, b^{2},\) and \(c^{2}\) equals the third.

Short answer

Diagonals are orthogonal if the sum of squares of two sides equals the square of the third.

Step-by-step solution

Understand the Problem

We need to show that a rectangular solid's diagonals are orthogonal if and only if the sum of the squares of two sides equals the square of the third side. A rectangular solid is essentially a 3D rectangle, having dimensions represented by lengths \(a\), \(b\), and \(c\). It has four space diagonals but we are interested in finding conditions for when two space diagonals are orthogonal.

Express the Space Diagonals

The space diagonal of a rectangular solid can be expressed by using the Pythagorean theorem in three dimensions. If one vertex of the rectangular solid is at the origin, then a diagonal's vector from this origin would be \( \langle a, b, c \rangle \). However, two diagonals starting from opposite ends could be represented by opposite vectors, i.e., \( \langle a, b, c \rangle \) and \( \langle -a, -b, c \rangle \).

Determine Orthogonality Condition

Two vectors are orthogonal if their dot product is zero. The dot product of vectors \(\langle a, b, c\rangle\) and \(\langle -a, -b, c\rangle\) is calculated as: \[ a(-a) + b(-b) + c(c) = -a^2 - b^2 + c^2 \]. Setting this equal to zero gives the equation \( -a^2 - b^2 + c^2 = 0 \), which simplifies to \( a^2 + b^2 = c^2 \).

Check Different Combinations

Similarly, considering combinations \( \langle a, b, -c \rangle \) and \( \langle -a, b, c \rangle \), we compute their dot product: \( -a^2 + b^2 - c^2 = 0 \), leading to \( a^2 = b^2 + c^2 \). Doing the same with \(a\) and \(b\) swapped, we derive \( b^2 = a^2 + c^2 \). Therefore, each condition represents a scenario where the sum of the squares of two sides is equal to the square of the third, verifying our hypothesis.

Conclusion

We have shown by checking the orthogonality condition and using the vector form of the diagonals, that if two space diagonals are orthogonal, then their side lengths must satisfy that the sum of two of \(a^2, b^2,\) and \(c^2\) equal the third. Hence, the statement is proven. Conversely, if any of the conditions \( a^2 + b^2 = c^2 \), \( a^2 = b^2 + c^2 \), or \( b^2 = a^2 + c^2 \) holds, two diagonals will indeed be orthogonal.

Key concepts

Pythagorean Theorem in Three Dimensions

The Pythagorean theorem is a fundamental concept traditionally applied in two dimensions. However, it extends naturally into three dimensions. In a rectangular solid, also known as a 3D rectangle, you have three perpendicular edges with lengths represented by \(a\), \(b\), and \(c\). Just like in a triangle, where the Pythagorean theorem states that the square of the hypotenuse equals the sum of the squares of the other two sides (\(a^2 + b^2 = c^2\)), in a rectangular solid, this can be extended to find the space diagonal.
The diagonal can be considered as a hypotenuse stretching across the solid. The length of this space diagonal can be calculated with:

• \(d = \sqrt{a^2 + b^2 + c^2}\)

This formula is essentially just an expansion of the Pythagorean theorem, taking the third dimension into account. In this context, it's crucial for determining properties like diagonal lengths and evaluating their orthogonality.

Vectors and Dot Product

Understanding vectors and their dot product is key to determining when two diagonals are orthogonal. A vector can be thought of as an arrow pointing from one point to another in space, and it has both direction and magnitude. In the case of a rectangular solid, we often describe a diagonal as a vector emanating from one vertex towards the opposite end. For example, the diagonal can be represented by the vector \( \langle a, b, c \rangle \).

The dot product is a mathematical operation that tells us how much one vector extends in the direction of another vector. If \( \mathbf{u} = \langle a, b, c \rangle \) and \( \mathbf{v} = \langle x, y, z \rangle \), their dot product is:

• \( \mathbf{u} \cdot \mathbf{v} = ax + by + cz \)

Vectors are orthogonal if their dot product equals zero. In our problem, we examine two diagonals: \( \langle a, b, c \rangle \) and \( \langle -a, -b, c \rangle \). Calculating their dot product:

• \( -a^2 - b^2 + c^2 = 0 \)

This indicates they are orthogonal if \( a^2 + b^2 = c^2 \). Similar calculations apply to other combinations, ensuring the observed properties.

Rectangular Solid Properties

Rectangular solids are simple but fascinating structures. They have properties that make them unique, especially when it comes to their geometry and spatial orientation. Understanding these properties helps in solving problems concerning their diagonals and other characteristics.

Key properties include:

• Six faces, each a rectangle.
• Twelve edges connecting the vertices.
• Four space diagonals connecting opposite corners.

Properties such as orthogonality of diagonals heavily rely on the geometric dimensions \(a\), \(b\), and \(c\). By the steps shown earlier, you can derive that two of the diagonals will be orthogonal if the dimensions satisfy properties derived from expanding the Pythagorean theorem.

This becomes particularly useful in practical applications where understanding the orientation and interaction of objects in 3D space is essential, such as in architecture and computer graphics. Being able to compute diagonal lengths and determine orthogonality directly impacts the structural integrity and visualization of designs.