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Chapter 11: Problem 42

Let \(\mathbf{u}\) and \(\mathbf{v}\) be adjacent sides of a parallelogram. Use vectors to prove that the parallelogram is a rectangle if the diagonals are equal in length.

Short Answer

Expert verified
The parallelogram is a rectangle if its diagonals are equal in length, as the sides are perpendicular.

Step by step solution

01

Vector Representation

Consider the vectors \( \mathbf{u} \) and \( \mathbf{v} \) as adjacent sides of the parallelogram. The diagonals \( \mathbf{d_1} \) and \( \mathbf{d_2} \) can be represented as vector sums: \( \mathbf{d_1} = \mathbf{u} + \mathbf{v} \) and \( \mathbf{d_2} = \mathbf{u} - \mathbf{v} \).
02

Set Diagonal Lengths Equal

Set the magnitudes of the diagonals equal, i.e. \( \| \mathbf{u} + \mathbf{v} \| = \| \mathbf{u} - \mathbf{v} \| \).
03

Use the Magnitude Formula

Using the magnitude formula, we expand the equation: \( \sqrt{(\mathbf{u} + \mathbf{v}) \cdot (\mathbf{u} + \mathbf{v})} = \sqrt{(\mathbf{u} - \mathbf{v}) \cdot (\mathbf{u} - \mathbf{v})} \).
04

Expand the Dot Products

The dot products yield: \( \mathbf{u} \cdot \mathbf{u} + 2\mathbf{u} \cdot \mathbf{v} + \mathbf{v} \cdot \mathbf{v} = \mathbf{u} \cdot \mathbf{u} - 2\mathbf{u} \cdot \mathbf{v} + \mathbf{v} \cdot \mathbf{v} \).
05

Simplify the Equation

Simplifying gives: \( 2\mathbf{u} \cdot \mathbf{v} = -2\mathbf{u} \cdot \mathbf{v} \).
06

Solution for Dot Product

This implies \( 4\mathbf{u} \cdot \mathbf{v} = 0 \). Therefore, \( \mathbf{u} \cdot \mathbf{v} = 0 \), which indicates that \( \mathbf{u} \) and \( \mathbf{v} \) are perpendicular.
07

Conclusion

Since the adjacent sides \( \mathbf{u} \) and \( \mathbf{v} \) are perpendicular, the parallelogram formed is a rectangle.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parallelogram
A parallelogram is a four-sided figure with opposite sides that are parallel and equal in length. When we talk about a parallelogram in vector calculus, we often represent its sides with vectors. These vectors help us calculate different properties, such as diagonal lengths, and determine whether a special case, like a rectangle, occurs.
  • The opposite angles of a parallelogram are equal.
  • The diagonals bisect each other.
  • A parallelogram remains unchanged if rotated by 180 degrees around its center.
Understanding these properties is crucial when solving vector problems involving parallelograms. They provide a framework for further exploration of shapes formed by vectors, leading us to identify specific conditions for various geometrical shapes within a parallelogram.
Vectors
Vectors are essential elements in this problem, representing sides of a parallelogram. A vector has both magnitude (length) and direction, often depicted as an arrow drawn from one point to another. In mathematical notation, vectors like \( \mathbf{u} \) and \( \mathbf{v} \) are used to define the sides of a parallelogram.
Key Points about Vectors:
  • Vectors can be added or subtracted to find resulting vectors.
  • They are written in terms of their components, such as \( \mathbf{u} = (u_1, u_2) \) and \( \mathbf{v} = (v_1, v_2) \).
  • Vector operations follow rules similar to algebraic equations but involve both direction and magnitude.
Understanding vectors and their operations lets us solve problems about geometric figures easily and precisely, like determining the properties of a parallelogram using its sides.
Diagonal Equality
In a parallelogram, the diagonals are segments connecting opposite corners. When these diagonals are equal in length, the parallelogram may exhibit special properties, such as forming a rectangle.
To investigate this, we must rely on the vectors representing the parallelogram's sides:
  • Vector expressions for diagonals: \( \mathbf{d_1} = \mathbf{u} + \mathbf{v} \) and \( \mathbf{d_2} = \mathbf{u} - \mathbf{v} \).
  • Diagonal lengths are equal if \( \|\mathbf{d_1}\| = \|\mathbf{d_2}\| \).
Checking diagonal equality using vector expressions can determine if the parallelogram is indeed a rectangle. This check leads directly to understanding conditions required for a rectangle's formation in vector geometry.
Dot Product
The dot product, also known as the scalar product, is a crucial concept in vector calculus. It provides a way to quantify the angle between two vectors, helping determine relationships like perpendicularity.
Calculate the dot product using the formula: \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \), where \( \mathbf{a} \) and \( \mathbf{b} \) are vectors.
  • A dot product of zero implies the vectors are perpendicular.
  • A positive dot product indicates both vectors point in roughly the same direction.
  • The dot product helps us solve problems involving diagonal equality by expanding equations to check perpendicular vector conditions.
Therefore, this tool is invaluable for deducing geometric properties like whether vectors form a rectangle when combined.
Perpendicular Vectors
Perpendicular vectors meet at a right angle, which geometrically confirms that their dot product equals zero. In the context of this problem, proving that two vectors \( \mathbf{u} \) and \( \mathbf{v} \) are perpendicular leads to the conclusion that the parallelogram is a rectangle.
Why are perpendicular vectors important?
  • They form the basis of right-angle triangles within vector spaces.
  • The perpendicularity condition (i.e., dot product equals zero) is critical for rectangles.
  • They simplify many calculations and proofs in vector geometry.
Knowing how to identify and work with perpendicular vectors enables us to explore and solve problems about geometric shapes efficiently, ensuring comprehension of compound shapes like rectangles within vector systems.

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