Problem 29 - Precalculus

Question

Determine whether each pair of vectors is orthogonal. $$\langle 0.8,4\rangle \text { and }\langle 3,-6\rangle$$

Step-by-Step Solution

Answer

The vectors are not orthogonal.

Step 1: Understand Orthogonality

Two vectors are orthogonal if their dot product is zero. Let's denote the two vectors as \( \mathbf{u} = \langle 0.8, 4 \rangle \) and \( \mathbf{v} = \langle 3, -6 \rangle \). Our task is to compute their dot product and check if it's zero.

Step 2: Calculate the Dot Product

The dot product of two vectors \( \mathbf{u} = \langle u_1, u_2 \rangle \) and \( \mathbf{v} = \langle v_1, v_2 \rangle \) is given by the formula \( u_1 \cdot v_1 + u_2 \cdot v_2 \). For our vectors, this means calculating \( 0.8 \times 3 + 4 \times (-6) \).

Step 3: Perform the Multiplication

Calculate each multiplication separately: 1. \( 0.8 \times 3 = 2.4 \)2. \( 4 \times (-6) = -24 \)

Step 4: Sum the Products

Add the results from Step 3: \( 2.4 + (-24) = 2.4 - 24 = -21.6 \). The dot product is \(-21.6\).

Step 5: Determine Orthogonality

Since the dot product is \(-21.6\), and not zero, the vectors \( \langle 0.8, 4 \rangle \) and \( \langle 3, -6 \rangle \) are not orthogonal.

Key Concepts

Dot Product

The dot product is a fundamental concept in vector mathematics. It allows us to operate on two vectors and get a scalar (a single number) as a result. The dot product is calculated by multiplying the corresponding components of the vectors and then summing the results. For two vectors \( \mathbf{u} = \langle u_1, u_2 \rangle \) and \( \mathbf{v} = \langle v_1, v_2 \rangle \), the dot product formula is: \[ u_1 \cdot v_1 + u_2 \cdot v_2 \] Breaking it down:

• Multiply the first components of each vector, \( u_1 \) and \( v_1 \).
• Multiply the second components of each vector, \( u_2 \) and \( v_2 \).
• Add these two results to get the dot product.

For example, if \( \mathbf{u} = \langle 0.8, 4 \rangle \) and \( \mathbf{v} = \langle 3, -6 \rangle \), the dot product is \( 0.8 \times 3 + 4 \times (-6) = 2.4 - 24 = -21.6 \). While dot products are used to find angles, orthogonality, and projections, it's crucial to understand this basic operation first.

Orthogonal Vectors

Orthogonal vectors are an exciting aspect of vector mathematics, representing vectors that meet at right angles. When two vectors are orthogonal, their dot product is zero. This stems from the geometric interpretation of the dot product, related to the cosine of the angle between the vectors. When this angle is 90 degrees, or \( \pi/2 \) radians, the cosine value is zero, making their dot product zero as well. This property is not just theoretical; it has practical applications in various fields:

• In physics, orthogonal vectors can represent forces acting perpendicularly, which do not influence each other directly.
• In computer graphics, orthogonal transformations maintain distances and angles, crucial for rendering accurate images.

To determine if vectors are orthogonal, simply compute their dot product. If it's zero, they are orthogonal. If not, like in our exercise with the vectors \( \langle 0.8, 4 \rangle \) and \( \langle 3, -6 \rangle \), which had a dot product of -21.6, they are not orthogonal.

Vector Mathematics

Vector mathematics extends beyond just dot products and orthogonality. Vectors are mathematical entities represented by arrows, having both magnitude (length) and direction. They are vital in expressing quantities that are dependent not only on amount but also on direction. Vectors can describe many real-world phenomena:

• In physics, vectors represent velocity, acceleration, and force, among others.
• In navigation, they are used to chart courses and determine position.

To work with vectors effectively, you need to understand operations like addition, subtraction, scalar multiplication, and the dot product. Each operation gives valuable insights and practical applications. For instance, the dot product helps determine angles between vectors, their projections, and orthogonality (as shown in our exercise). Understanding vector mathematics equips you with tools to analyze diverse data and solve complex problems in science and engineering.