Problem 80 - A Graphical Approach to Precalculus with Limits

Question

Determine whether each pair of vectors is orthogonal. $$\langle 3,4\rangle,\langle 6,8\rangle$$

Step-by-Step Solution

Answer

The vectors are not orthogonal.

Step 1: Understand Orthogonality

Two vectors are orthogonal if their dot product is zero. The dot product for vectors \( \langle a, b \rangle \) and \( \langle c, d \rangle \) is calculated as \( a \cdot c + b \cdot d \).

Step 2: Calculate the Dot Product

Given vectors are \( \langle 3, 4 \rangle \) and \( \langle 6, 8 \rangle \). Calculate the dot product: \( 3 \cdot 6 + 4 \cdot 8 = 18 + 32 = 50 \).

Step 3: Conclusion: Determine Orthogonality

Since the dot product is 50, which is not zero, the vectors are not orthogonal.

Key Concepts

Dot Product

The dot product is a crucial concept when evaluating the relationship between two vectors, especially in determining whether they are orthogonal. The dot product is calculated by multiplying corresponding components of two vectors and then adding the results. Mathematically, the dot product of two vectors, \( \langle a, b \rangle \) and \( \langle c, d \rangle \), is defined as:

• \( a \cdot c + b \cdot d \)

This mathematical operation is foundational in precalculus and extends into many applications in physics and engineering, helping us to determine angles between vectors and work done by forces, among other things.

Vectors

Vectors are essential building blocks in mathematics that have both magnitude (length) and direction. In precalculus, we often work with two-dimensional vectors represented as ordered pairs, such as \( \langle 3, 4 \rangle \), which consist of two components.

• The first number is the horizontal component, which shows how far along the x-axis the vector reaches.
• The second number is the vertical component, indicating the vector's reach along the y-axis.

Understanding vectors allows you to visualize and solve problems involving quantities that have both size and direction, like force, velocity, and displacement.

Orthogonality

Orthogonality is a key idea in vector mathematics and refers to the perpendicularity of two vectors. Two vectors are orthogonal if they form a 90-degree angle with each other, which is often described as being perpendicular in geometric terms.
In algebraic terms, this means that their dot product is zero. If the dot product of two vectors is zero, it signifies that they are orthogonal. This property has practical applications in fields like computer graphics, where determining whether lines or planes are perpendicular is vital for rendering and design.

Precalculus

Precalculus serves as a preparatory course for calculus, incorporating a wide range of mathematical concepts including vectors and their operations. Key topics like the dot product and orthogonality are covered to provide a solid foundation for more advanced studies in calculus.
Precalculus not only deals with understanding mathematical properties and calculations but also enhances problem-solving skills. Students who grasp these concepts are better equipped to tackle calculus concepts such as derivatives and integrals, which rely on a strong understanding of vector manipulation and properties.