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Chapter 1: Problem 11

Recall that a rhombus is a planar quadrilateral whose sides all have the same length. Use the dot product to show that the diagonals of a rhombus are perpendicular to each other.

Short Answer

Expert verified
The diagonals of a rhombus are perpendicular because their dot product equals zero.

Step by step solution

01

Define the Rhombus Vertices

Consider a rhombus with vertices \( A(0,0) \), \( B(a,0) \), \( C(a+b_h,b_v) \), and \( D(b_h,b_v) \). This setup uses vectors to define each point, ensuring all sides have equal length.
02

Verify Equal Side Lengths

Compute the distances for each side using the distance formula: \( AB \), \( BC \), \( CD \), and \( DA \). Ensure each distance is \( a \) to confirm it is a rhombus.
03

Determine Vector Representations

Write the vectors for the diagonals: \( AC = (a+b_h, b_v) \) and \( BD = (b_h-a, b_v) \). These represent the rhombus's diagonals.
04

Calculate the Dot Product of Diagonals

Find the dot product of vectors \( AC \) and \( BD \):\[(a + b_h, b_v) \cdot (b_h - a, b_v) = (a + b_h) \cdot (b_h - a) + b_v^2 = a b_h - a^2 + b_h b_h - b_h a + b_v^2\]Simplify to find if the dot product equals zero.
05

Show Perpendicularity

For the diagonals to be perpendicular, the dot product must be zero. Simplify:\[-a^2 + b_h^2 + b_v^2 = 0 \]Since this equation equals zero, the diagonals of the rhombus are perpendicular.

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Key Concepts

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

Dot Product
The dot product is a way to multiply two vectors together, resulting in a scalar value. It's used to determine the angle or relationship between two vectors. If the dot product is zero, the vectors are perpendicular. For two vectors \((\mathbf{u})\) and \((\mathbf{v})\), the dot product is calculated as:\[\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + \ldots + u_nv_n\]
  • Each term pairs corresponding components from the vectors.
  • The result is a single number.
In geometrical terms, the dot product can also be given by:\[\mathbf{u} \cdot \mathbf{v} = ||\mathbf{u}|| ||\mathbf{v}|| \cos(\theta)\]Where \( \theta \) is the angle between the vectors. A zero dot product implies \( \theta = 90^\circ \), or that the vectors are perpendicular.
Quadrilateral
A quadrilateral is any four-sided shape in a plane. In geometry, quadrilaterals can take many forms, such as squares, rectangles, parallelograms, and rhombuses. Each quadrilateral has:
  • Four vertices
  • Four sides
  • Two diagonals
Different types of quadrilaterals are defined by unique properties like equal side lengths or parallel opposite sides. The rhombus, for example, is a special type where all side lengths are equal. Understanding quadrilaterals is essential for grasping more complex geometrical concepts.
Diagonals Perpendicular
Diagonals intersecting at right angles (perpendicular diagonals) create significant geometric properties. In a rhombus, this characteristic arises naturally due to the figure's symmetry.When the diagonals of a quadrilateral are perpendicular, it implies that their intersecting angle is \(90^\circ\). This can be verified using the dot product, as explored in the solution to our exercise:
  • Calculate the dot product of the diagonals.
  • If the dot product equals zero, the diagonals are indeed perpendicular.
This is crucial in many contexts, especially when defining special properties or types of quadrilaterals, like rhombuses where this perpendicularity is intrinsic.
Equal Side Lengths
Equal side lengths mean that each side of a shape has the same measure. For a quadrilateral like a rhombus, this is a defining property. All four sides being the same length ensures certain symmetry and geometrical properties.To confirm equal lengths, the distance between vertices can be calculated using the distance formula:\[d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\]
  • Apply this formula for each side of the quadrilateral.
  • Verify that all calculated distances are equal.
In the case of a rhombus, this criterion distinguishes it from other quadrilaterals and validates its unique characteristics, such as perpendicular diagonals.

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Most popular questions from this chapter

We have seen that the commutative law \(A B=B A\) does not hold in general for matrix multiplication. In fact, the situation is worse than that: it's rare for \(A B\) and \(B A\) even to both be defined. In that sense, the preceding half dozen exercises are misleading. (a) Find an example of matrices \(A\) and \(B\) such that neither \(A B\) nor \(B A\) is defined. (b) Find an example where \(A B\) is defined but \(B A\) is not.

Show that the dot product satisfies the given property. The properties are true for vectors in \(\mathbb{R}^{n},\) though you may assume in your arguments that the vectors are in \(\mathbb{R}^{2}\), i.e., \(\mathbf{x}=\left(x_{1}, x_{2}\right), \mathbf{y}=\left(y_{1}, y_{2}\right),\) and so on. The proofs for \(\mathbb{R}^{n}\) in general are similar. \(\|c \mathbf{x}\|=|c|\|\mathbf{x}\|\) for any scalar \(c\)

Find the given determinant. Find the area of the parallelogram in \(\mathbb{R}^{2}\) determined by \(\mathrm{x}=(-2,-3)\) and \(\mathrm{y}=(-3,2)\).

Let \(T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) be the linear transformation such that \(T\left(\mathbf{e}_{1}\right)=(1,2)\) and \(T\left(\mathbf{e}_{2}\right)=(3,4)\). (a) Find the matrix of \(T\) with respect to the standard bases. (b) Find \(T(5,6)\). (c) Find a general formula for \(T\left(x_{1}, x_{2}\right)\).

In Exercises \(6.1-6.4\), find the given determinant. \(\operatorname{det}\left[\begin{array}{rr}2 & 5 \\ -3 & 4\end{array}\right]\)
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