Problem 4 Interpret these results geometri... [FREE SOLUTION]

Chapter 1: Problem 4

Interpret these results geometrically in terms of the parallelogram formed by $x$ and $y$. Verify the Cauchy-Schwarz inequality and the triangle inequality for the vectors. $$\mathbf{x}=(1,0,2,6), \mathbf{y}=(3,8,4,1)$$

Short Answer

Expert verified

Cauchy-Schwarz and triangle inequalities are both satisfied for vectors $\mathbf{x}$ and $\mathbf{y}$.

Step by step solution

Calculate the Dot Product

To verify the Cauchy-Schwarz inequality, we first need the dot product of $ \mathbf{x} $ and $ \mathbf{y} $: $ \mathbf{x} \cdot \mathbf{y} = (1)(3) + (0)(8) + (2)(4) + (6)(1) = 3 + 0 + 8 + 6 = 17 $.

Calculate the Magnitudes

Next, calculate the magnitudes of both vectors. For $ \mathbf{x} $: $ ||\mathbf{x}|| = \sqrt{1^2 + 0^2 + 2^2 + 6^2} = \sqrt{1 + 0 + 4 + 36} = \sqrt{41} $.For $ \mathbf{y} $: $ ||\mathbf{y}|| = \sqrt{3^2 + 8^2 + 4^2 + 1^2} = \sqrt{9 + 64 + 16 + 1} = \sqrt{90} $.

Verify Cauchy-Schwarz Inequality

The Cauchy-Schwarz inequality states that $ |\mathbf{x} \cdot \mathbf{y}| \leq ||\mathbf{x}|| \times ||\mathbf{y}|| $. We calculate:$ ||\mathbf{x}|| \times ||\mathbf{y}|| = \sqrt{41} \times \sqrt{90} = \sqrt{3690} $.Check the inequality: $ 17 \leq \sqrt{3690} $. Since 17 is less than $ \sqrt{3690} $, the inequality holds.

Calculate Vector Sums

For the triangle inequality, compute the sum of magnitudes:$ ||\mathbf{x} + \mathbf{y}|| $ First, find the vector sum: $ \mathbf{x} + \mathbf{y} = (1+3, 0+8, 2+4, 6+1) = (4, 8, 6, 7) $.Find the magnitude: $ ||\mathbf{x} + \mathbf{y}|| = \sqrt{4^2 + 8^2 + 6^2 + 7^2} = \sqrt{16 + 64 + 36 + 49} = \sqrt{165} $.

Verify Triangle Inequality

The triangle inequality states that:$ ||\mathbf{x} + \mathbf{y}|| \leq ||\mathbf{x}|| + ||\mathbf{y}|| $.Calculate the right term:$ ||\mathbf{x}|| + ||\mathbf{y}|| = \sqrt{41} + \sqrt{90} $. The condition is verified if:$ \sqrt{165} \leq \sqrt{41} + \sqrt{90} $. Both sides are positive, confirming that the triangle inequality holds.

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Key Concepts

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

triangle inequality

The triangle inequality is a fundamental concept in vector analysis. It states that for any vectors $ \mathbf{x} $ and $ \mathbf{y} $, the magnitude of their sum is less than or equal to the sum of their magnitudes:
$ ||\mathbf{x} + \mathbf{y}|| \leq ||\mathbf{x}|| + ||\mathbf{y}|| $.
Imagine two sides of a triangle represented by vectors $ \mathbf{x} $ and $ \mathbf{y} $. The vector sum $ \mathbf{x} + \mathbf{y} $, which is the third side, cannot be longer than the sum of the other two sides.

This lets us visually imagine the concept:

The sum of the lengths is always stretched to its maximum when vectors are aligned.
In other cases, the added vectors form angles, shortening the vector sum relative to the direct path.

This inequality roots from geometry and ensures that vectors mimic the properties of triangles in mathematical spaces.

vector dot product

The vector dot product is a way to quantify the relationship between two vectors. For two vectors $ \mathbf{x} = (x_1, x_2, ..., x_n) $ and $ \mathbf{y} = (y_1, y_2, ..., y_n) $, the dot product is calculated as:
$ \mathbf{x} \cdot \mathbf{y} = x_1 y_1 + x_2 y_2 + ... + x_n y_n $.
This product results in a scalar and serves several purposes in vector mathematics.

Here are a few key points about the dot product:

It measures how two vectors align with each other.
When the dot product is zero, the vectors are orthogonal (perpendicular).
The larger the dot product, the more parallel the vectors are.

This scalar is crucial in confirming inequalities such as Cauchy-Schwarz, which impacts many areas of physics and engineering by showing vector relations.

vector magnitudes

The magnitude of a vector is akin to finding the length of a straight line segment in Euclidean space, fundamentally rooted in the Pythagorean theorem. For a vector $ \mathbf{x} = (x_1, x_2, ..., x_n) $, its magnitude, denoted as $ ||\mathbf{x}|| $, is given by:
$ ||\mathbf{x}|| = \sqrt{x_1^2 + x_2^2 + ... + x_n^2} $.

This calculation draws a clear image of the vector's size or length in its space.

Magnitudes tell us about the distance a vector covers from the origin to its endpoint.
It provides foundational elements for evaluating and verifying inequalities like the Cauchy-Schwarz inequality.
A zero magnitude indicates the absence or zero existence of the vector in that space.

Understanding magnitudes is critical to visualizing the geometry of vector spaces.

parallelogram interpretation

The parallelogram interpretation provides a visual way to understand operations involving vectors, especially sums, dot products, and magnitudes. Imagine vectors $ \mathbf{x} $ and $ \mathbf{y} $ forming adjacent sides of a parallelogram. The sum of these vectors, $ \mathbf{x} + \mathbf{y} $, creates the diagonal across the parallelogram.

In this geometric framework:

The diagonal length reflects the vector sum magnitude.
The parallelogram helps visualize how vectors add up spatially.
The area can relate to the dot product magnitude when vectors are orthogonal.

Understanding this concept gives clearer insight into complex vector operations and provides a tangible interpretation of abstract vector mathematics.

91影视

Interpret these results geometrically in terms of the parallelogram formed by \(x\) and \(y\). Verify the Cauchy-Schwarz inequality and the triangle inequality for the vectors. $$\mathbf{x}=(1,0,2,6), \mathbf{y}=(3,8,4,1)$$

Short Answer

Step by step solution

Calculate the Dot Product

Calculate the Magnitudes

Verify Cauchy-Schwarz Inequality

Calculate Vector Sums

Verify Triangle Inequality

Key Concepts

triangle inequality

vector dot product

vector magnitudes

parallelogram interpretation

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