Problem 21 - University Calculus: Early Transcendentals

Question

Show that the diagonals of a rhombus (parallelogram with sides of equal length) are perpendicular.

Step-by-Step Solution

Answer

The diagonals of a rhombus are perpendicular because their dot product is zero, confirming they intersect at right angles.

Step 1: Understanding the Problem

A rhombus is a quadrilateral where all four sides have equal length. One of the properties we need to prove is that the diagonals are perpendicular, meaning they intersect at right angles.

Step 2: Identify Known Properties of Rhombus

Recall that the diagonals of a rhombus bisect each other. This means that each diagonal cuts the other into two equal parts.

Step 3: Using Vector Representation

Consider the rhombus as vectors. Let the vectors for the sides be \( \vec{a} \) and \( \vec{b} \). Then, the diagonal vectors can be represented as \( \vec{a} + \vec{b} \) and \( \vec{a} - \vec{b} \).

Step 4: Calculate the Dot Product of Diagonals

To prove that the diagonals are perpendicular, their dot product should be zero. Calculate: \[ (\vec{a} + \vec{b}) \cdot (\vec{a} - \vec{b}) = \vec{a} \cdot \vec{a} - \vec{b} \cdot \vec{b} \] Since \( \vec{a} \cdot \vec{b} = 0 \) in perpendicular vectors configuration, concentrate on \( \vec{a} \cdot \vec{a} - \vec{b} \cdot \vec{b} \).

Step 5: Apply Equal Magnitude of Sides

Since \( \vec{a} \) and \( \vec{b} \) are of equal magnitude, we have \( \|\vec{a}\| = \|\vec{b}\| \). Thus, \[ \vec{a} \cdot \vec{a} = \vec{b} \cdot \vec{b} \] Resulting in: \[ \vec{a} \cdot \vec{a} - \vec{b} \cdot \vec{b} = 0 \].

Step 6: Conclusion - Perpendicular Diagonals

Since the dot product is zero, the diagonals \( \vec{a} + \vec{b} \) and \( \vec{a} - \vec{b} \) are perpendicular. Therefore, the diagonals of a rhombus are always perpendicular.

Key Concepts

Diagonals of a Rhombus

A rhombus is a special type of parallelogram where all four sides are equal in length. But what makes its diagonals so interesting? First, remember that each diagonal of a rhombus bisects the other, meaning they cut each other exactly in half. This is a distinguishing feature of the rhombus shape. Now, when we say the diagonals are perpendicular, it means when they intersect, they do so at a right angle (90 degrees). It is one of the classic properties that make the rhombus unique among quadrilaterals. When you examine a rhombus, its opposite angles are equal and its consecutive angles are supplementary. These properties all tie back to the significance of how the diagonals interact. By using this property, you can determine many things about the rhombus, such as area calculations which involve half the product of its diagonals.

Vector Representation

Vectors are a powerful way to understand geometry and are especially useful in proving properties like those of a rhombus' diagonals. To represent the sides of the rhombus as vectors, consider vectors \( \vec{a} \) and \( \vec{b} \). These vectors are not just arbitrary; they correspond to the sides of the rhombus. The diagonal vectors are formed by combining these side vectors. Specifically, the diagonal vectors can be expressed as \( \vec{a} + \vec{b} \) and \( \vec{a} - \vec{b} \). By using vectors, you can visualize and analyze the directions and lengths of the diagonals relative to the sides, which helps in exploring the geometric properties of the rhombus. Vectors simplify the task of proving attributes like the perpendicularity of the diagonals, allowing a mathematical approach that is grounded in both algebraic and geometric principles.

Dot Product

The dot product is a mathematical operation that takes two vectors and returns a scalar (a single number). It plays a crucial role in determining the angle between two vectors. When the dot product of two vectors is zero, it indicates that the vectors are perpendicular as they meet at a right angle. For a rhombus, the key step is to calculate the dot product of the diagonals \( (\vec{a} + \vec{b}) \cdot (\vec{a} - \vec{b}) \). This calculation involves expanding the expression to get \( \vec{a} \cdot \vec{a} - \vec{b} \cdot \vec{b} \) due to the property of dot products being distributive. When vectors \( \vec{a} \) and \( \vec{b} \) have equal magnitude (same length), it simplifies to zero, thus verifying that the diagonals are perpendicular. Using the dot product in this context allows us to confirm geometric properties algebraically.

Perpendicular Diagonals

Perpendicular diagonals mean that the diagonals cut each other at a 90-degree angle. For the rhombus, this is not just a theoretical property but a practical one, mathematically shown using vectors and the dot product. Once you have established that \( (\vec{a} + \vec{b}) \) dot \( (\vec{a} - \vec{b}) = 0 \), you have proven the diagonals are perpendicular. This perpendicularity tells you that the rhombus can be divided into four right-angle triangles. Each triangle has the diagonals as its legs, emphasizing the symmetry and balance of the rhombus' structure. This property is helpful in various applications, such as calculating areas and analyzing forces in physics, as it simplifies complex situations into manageable right-angle components. By understanding why the diagonals are perpendicular, you gain deeper insights into the geometric beauty and functional utility of the rhombus.