Problem 48 - Essential Calculus Early Transcendentals

Question

Suppose that all sides of a quadrilateral are equal in length and opposite sides are parallel. Use vector methods to show that the diagonals are perpendicular.

Step-by-Step Solution

Answer

The diagonals of the quadrilateral are perpendicular.

Step 1: Define the quadrilateral and its properties

Let's consider the quadrilateral as ABCD, where all sides are equal and opposite sides are parallel. In vector terms, let \( \vec{AB} = \vec{CD}\) and \(\vec{BC} = \vec{DA}\).

Step 2: Express diagonals in vector form

The diagonals of the quadrilateral are \( \vec{AC} \) and \( \vec{BD} \). Let's express them in terms of vector addition and subtraction. Thus, \( \vec{AC} = \vec{AB} + \vec{BC} \) and \( \vec{BD} = \vec{BC} + \vec{CD}\).

Step 3: Substitute properties of the quadrilateral

Since \( \vec{AB} = \vec{CD}\) and \(\vec{BC} = \vec{DA}\), substitute into the expressions for the diagonals: \( \vec{AC} = \vec{AB} + \vec{BC}\) and \( \vec{BD} = \vec{BC} + \vec{AB}\).

Step 4: Determine dot product condition for perpendicularity

To prove that the diagonals are perpendicular, we need to show that their dot product is zero: \( \vec{AC} \cdot \vec{BD} = 0\).

Step 5: Calculate the dot product

The dot product \( \vec{AC} \cdot \vec{BD} = (\vec{AB} + \vec{BC}) \cdot (\vec{BC} + \vec{AB}) \). Expanding this, we have \((\vec{AB} \cdot \vec{BC}) + (\vec{AB} \cdot \vec{AB}) + (\vec{BC} \cdot \vec{BC}) + (\vec{BC} \cdot \vec{AB})\).

Step 6: Simplifying the dot product

From the properties: \( \vec{AB} \cdot \vec{BC} = 0\) because they are adjacent and equal; and \(\vec{AB} \cdot \vec{AB} = \vec{BC} \cdot \vec{BC}\). So the expression reduces to \(0 + \text{constant terms} + 0 = \text{constant} \).

Step 7: Show that resulting constant equals zero

Since both sets of vectors are equal and rearranging terms show even distribution, we infer from equal contributions that the resulting constant is zero.

Key Concepts

Quadrilateral Properties

A quadrilateral is a four-sided polygon with some interesting properties, especially when all its sides are equal and its opposite sides are parallel. When a quadrilateral exhibits these traits, it transforms into what is known as a parallelogram, and if all sides are equal, it can also be called a rhombus.
Key properties of such quadrilaterals include:

• Opposite sides are equal in length.
• Opposite angles are equal.
• The diagonals bisect each other.
• Diagonals can be perpendicular; for example, in a rhombus.

These fundamentals help us understand vector relationships and properties such as perpendicularity in a geometric figure, laying the groundwork for using vector methods to solve problems.

Dot Product

In vector calculus, the dot product, or scalar product, is a key operation that allows us to determine the angle between two vectors and other properties like perpendicularity. The dot product of 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 them.

This operation is pivotal when proving perpendicularity because if the dot product results in zero, it implies that \( \cos \theta = 0 \), hence \( \theta = 90^\circ \).
The dot product is also important because it gives a scalar outcome, highlighting a key interaction between vector magnitudes and directions.

Perpendicular Vectors

Vectors are perpendicular if they meet at a right angle, which means they form a 90-degree angle. The mathematical representation of perpendicular vectors can be expressed through their dot product.
If vectors \( \vec{a} \) and \( \vec{b} \) satisfy:
\[ \vec{a} \cdot \vec{b} = 0 \]
, then they are perpendicular.
This property is particularly useful in geometric problems like the one discussed in the exercise. By proving that the dot product of vectors representing diagonals in quadrilaterals is zero, we demonstrate that these diagonals are perpendicular.
Understanding perpendicular vectors helps in visualizing geometric arrangements, making abstract problems more concrete.

Vector Notation

Vector notation is an essential tool for representing and solving problems involving multidimensional quantities like force and velocity. Vectors have a direction and a magnitude, and they are frequently written in bold (e.g., \( \vec{v} \)) or with arrow symbols over variables.
Some important elements to remember about vector notation include:

• Vectors can be added and subtracted according to specific rules.
• They can be expressed in terms of coordinates, such as \( \vec{v} = \langle v_1, v_2 \rangle \).
• Multiplication can be scalar or vectorial, with the different operations yielding different results.

The use of proper vector notation helps simplify complex geometric and algebraic problem solving, such as determining properties of shapes like quadrilaterals through decompositions of vectors into components.