Problem 25 - Precalculus

Question

Determine whether each pair of vectors is orthogonal. $$\langle-6,8\rangle \text { and }\langle-8,6\rangle$$

Step-by-Step Solution

Answer

The vectors are not orthogonal because their dot product is 96, not zero.

Step 1: Understand the Concept of Orthogonal Vectors

Two vectors are said to be orthogonal if their dot product is zero. In terms of the vectors \( \langle a, b \rangle \) and \( \langle c, d \rangle \), the dot product is calculated as \( a \cdot c + b \cdot d \). To determine if vectors are orthogonal, we need to find this dot product and check if it equals zero.

Step 2: Compute the Dot Product

Given vectors \( \langle -6, 8 \rangle \) and \( \langle -8, 6 \rangle \), their dot product is calculated as:\[(-6) \cdot (-8) + 8 \cdot 6\]Expand the products to obtain:\[48 + 48\]

Step 3: Verify if the Dot Product is Zero

After computing the dot product, we have:\[48 + 48 = 96\]The dot product is 96, which is not equal to zero, hence the vectors are not orthogonal.

Key Concepts

Dot Product

The dot product is a key operation in vector mathematics that helps us determine relationships between vectors, such as parallelism and orthogonality. Calculating the dot product involves multiplying the corresponding components of two vectors and then summing those products. Given vectors \( \langle a, b \rangle \) and \( \langle c, d \rangle \), the dot product is expressed as:

• \( a \cdot c + b \cdot d \)

If the final result of the dot product calculation is zero, the vectors are perpendicular (orthogonal). In the case of our vectors \( \langle-6,8\rangle \) and \( \langle-8,6\rangle \), the dot product results in a value of 96, which indicates they are not orthogonal as this result is not zero.
Understanding this simple calculation can reveal much about the orientation and relationship between vectors.

Vector Operations

Vector operations are fundamental in mathematics and physics for representing quantities with both magnitude and direction. One common operation is finding the sum, which involves adding corresponding components of vectors, or in the case of the dot product, finding relationships between vectors. Understanding how to interpret and calculate with vectors offers insights into these geometric entities. It includes operations like:

• Addition: Adding the corresponding components of two vectors.
• Subtraction: Subtracting corresponding components of vectors.
• Scalar multiplication: Multiplying a vector by a scalar, affecting its magnitude.
• Dot Product: A unique operation for determining orthogonality or projection, as previously explained.

These operations allow manipulation of vectors to model real-world scenarios such as forces and velocities, among many others.

Precalculus Concepts

Precalculus helps lay the groundwork for understanding more complex mathematical ideas, preparing students for calculus. It encompasses a wide array of topics including the study of functions, complex numbers, and importantly, vectors. Vectors are essential in precalculus because they introduce students to mathematical objects that have both magnitude and direction, rather than just numerical values. Important concepts include:

• Introduction to vectors: Understanding vectors' components and their geometric representation.
• Dot product: Recognizing it as a tool for vector analysis.
• Magnitude of a vector: Calculating the length of a vector using the Pythagorean theorem.

These components are crucial, not just for problem solving, but for developing analytical thinking and the ability to model complex systems, skills that are vital in calculus and other advanced mathematics.