Problem 81 - A Graphical Approach to Precalculus with Limits

Question

Determine whether each pair of vectors is orthogonal. $$\langle 1,0\rangle,\langle\sqrt{2}, 0\rangle$$

Step-by-Step Solution

Answer

The vectors are not orthogonal.

Step 1: Understand the Criteria for Orthogonality

Two vectors are orthogonal if their dot product is equal to zero. The dot product of two vectors \( \langle a, b \rangle \) and \( \langle c, d \rangle \) is calculated as \( a \times c + b \times d \).

Step 2: Calculate the Dot Product

Now calculate the dot product of the given vectors \( \langle 1, 0 \rangle \) and \( \langle \sqrt{2}, 0 \rangle \). This calculation is: \[1 \times \sqrt{2} + 0 \times 0 = \sqrt{2}\]

Step 3: Determine Orthogonality

The result of the dot product in Step 2 is \( \sqrt{2} \). Since \( \sqrt{2} eq 0 \), the vectors are not orthogonal.

Key Concepts

Orthogonality

Orthogonality of vectors is a crucial concept in vector algebra. Two vectors are said to be orthogonal when they meet at a 90-degree angle. Geometrically, this means the vectors are perpendicular to each other. Orthogonality is not just a geometric concept but can also be determined algebraically using the dot product. If the dot product of two vectors is zero, the vectors are orthogonal.

Orthogonality is important because it helps simplify problems in physics and engineering. For example, in linear equations or when analyzing forces, knowing that two components are orthogonal helps decompose them into simpler parts along different axes. This is why understanding and calculating orthogonality can be a powerful tool in various fields.

In summary, orthogonality is about answering whether vectors cross paths at a right angle. It's a visual and analytical check on how vectors relate to one another in space.

Dot Product

The dot product is a way to multiply two vectors, yielding a scalar result. This scalar is a measure of how much one vector extends in the direction of another. To find the dot product of two vectors, you need to multiply their corresponding components and then sum those products.

Given vectors \( \langle a, b \rangle \) and \( \langle c, d \rangle \), the dot product is calculated as:

• \( a \times c + b \times d \)

This calculation provides a single numerical value.

An important property of the dot product is that it provides a straightforward check for orthogonality. If the result of a dot product is zero, the vectors are orthogonal. For example, in the provided exercise, the vectors \( \langle 1, 0 \rangle \) and \( \langle \sqrt{2}, 0 \rangle \) were not orthogonal because their dot product was \( \sqrt{2} \), not zero.

Understanding the dot product is essential because it links vector algebra to geometry, helping us visualize and evaluate spatial relationships between vectors.

Vector Calculation

Vector calculation involves operations like addition, subtraction, and scalar multiplication, but a vital operation is the dot product, which helps determine orthogonality.

When performing vector calculations, it is important to understand the components of a vector. Each vector in a two-dimensional plane is represented by two components, such as \( \langle a, b \rangle \), representing the x and y-axis directions.

To determine the orthogonality of vectors, vector calculation often focuses on the dot product:

• Multiply corresponding components of both vectors.
• Sum these products.
• Evaluate the result. If zero, the vectors are orthogonal.

In more complex scenarios, vectors can operate in three-dimensional space with three components. Thus, understanding these operations is important not only in math but also in fields like physics and computer graphics where vectors are used to model reality. By mastering vector calculations, you empower yourself to solve a wide range of problems where direction and magnitude play crucial roles.