Problem 33 - Precalculus

Question

Determine whether each pair of vectors is orthogonal. $$\langle\sqrt{3}, \sqrt{6}\rangle \text { and }\langle-\sqrt{2}, 1\rangle$$

Step-by-Step Solution

Answer

The vectors are orthogonal.

Step 1: Understand the Concept of Orthogonality

Two vectors are orthogonal if their dot product is equal to zero. Therefore, if the dot product of the given vectors \(\langle\sqrt{3}, \sqrt{6}\rangle\) and \(\langle-\sqrt{2}, 1\rangle\) is equal to zero, they are orthogonal.

Step 2: Calculate the Dot Product

The dot product of two vectors \(\langle a, b \rangle\) and \(\langle c, d \rangle\) is calculated as follows: \(a \times c + b \times d\). For the vectors given: \(\langle \sqrt{3}, \sqrt{6} \rangle\) and \(\langle -\sqrt{2}, 1 \rangle\), compute the dot product: \(\sqrt{3} \cdot (-\sqrt{2}) + \sqrt{6} \cdot 1\).

Step 3: Simplify the Dot Product Calculation

First, calculate \(\sqrt{3} \cdot (-\sqrt{2}) = -\sqrt{6}\). Then, calculate \(\sqrt{6} \cdot 1 = \sqrt{6}\). Combine the results to get the total dot product: \(-\sqrt{6} + \sqrt{6}\).

Step 4: Determine if the Dot Product is Zero

Calculate \(-\sqrt{6} + \sqrt{6} = 0\). Since the dot product is zero, the vectors are orthogonal.

Key Concepts

Understanding the Dot Product

The dot product is a fundamental operation in vector algebra. It is used to find the scalar product of two vectors. The dot product of vectors \( \langle a, b \rangle \) and \( \langle c, d \rangle \) is calculated as \( a \times c + b \times d \). This operation is particularly useful in determining whether vectors are orthogonal.

- If the result of the dot product is zero, the vectors are orthogonal (perpendicular to each other). - This means there's no projection of one vector onto the other, indicating a 90-degree angle between them.

In practical terms, like our example with vectors \( \langle\sqrt{3}, \sqrt{6}\rangle \) and \( \langle-\sqrt{2}, 1\rangle \), we apply the dot product formula. Calculating \( \sqrt{3} \cdot (-\sqrt{2}) + \sqrt{6} \cdot 1 \) shows how algebra simplifies the connection between vectors.

Vector Algebra Basics

Vector algebra involves operations and properties of vectors, which are quantities expressed with both magnitude and direction. Understanding vector algebra is crucial for solving a variety of problems in physics, engineering, and mathematics.

Key operations include:

• Addition and Subtraction: Combining vectors by adding or subtracting their components.
• Scalar Multiplication: Multiplying a vector by a numerical value, affecting its magnitude but not its direction.
• Dot Product: As discussed, results in a scalar and helps us find orthogonal vectors.
• Cross Product: Results in another vector that is perpendicular to two given vectors in 3 dimensions.

In our example, using the dot product is a straightforward way to verify orthogonality, helping us see if two vectors meet at a right angle, like \( \langle\sqrt{3}, \sqrt{6}\rangle\) and \( \langle-\sqrt{2}, 1\rangle\).

Precalculus and Vector Applications

Precalculus prepares students for the concepts in calculus and beyond, including a solid grasp of vectors. While dealing with precalculus, understanding vector operations like dot products and other algebraic concepts is vital.

Precalculus covers:

• Functions and Graphs: Understanding the behavior of different types of functions, including vector functions.
• Matrices and Determinants: Tools for solving systems of equations, crucial in vector operations.
• Trigonometry: Essential for understanding angles and vector directions.

The role of vectors is significant as they are used in various precalculus problems, helping build an analytical foundation. The exercise we tackled about vector orthogonality is a stepping stone for comprehending more advanced mathematical concepts, showing the importance of dot products as part of a comprehensive precalculus curriculum.