Problem 31 - Precalculus

Question

Determine whether each pair of vectors is orthogonal. $$\langle 5,-0.4\rangle \text { and }\langle 1.6,20\rangle$$

Step-by-Step Solution

Answer

The vectors are orthogonal because their dot product is 0.

Step 1: Understand Orthogonal Vectors

Two vectors are orthogonal if their dot product is zero. In this problem, we need to check the dot product of the vectors \( \langle 5, -0.4 \rangle \) and \( \langle 1.6, 20 \rangle \).

Step 2: Calculate the Dot Product

To find the dot product of two vectors \( \langle a, b \rangle \) and \( \langle c, d \rangle \), use the formula: \[ a \cdot c + b \cdot d \]. For our vectors, compute \( 5 \times 1.6 + (-0.4) \times 20 \).

Step 3: Perform the Calculations

Calculate each part of the dot product:1. Calculate \( 5 \times 1.6 = 8 \).2. Calculate \( -0.4 \times 20 = -8 \).3. Add the results: \( 8 + (-8) = 0 \).

Step 4: Conclusion

Since the dot product is zero, \( 8 + (-8) = 0 \), the vectors \( \langle 5, -0.4 \rangle \) and \( \langle 1.6, 20 \rangle \) are orthogonal.

Key Concepts

dot product

The dot product is a fundamental concept in vector algebra where two vectors are multiplied together to produce a scalar. This operation helps determine whether two vectors are orthogonal, meaning they are perpendicular in space. You calculate the dot product using the formula:

• For vectors \( \langle a, b \rangle \) and \( \langle c, d \rangle \), the dot product is \( a \cdot c + b \cdot d \).

Let's break it down with simple steps: Multiply the corresponding components of each vector, and then add those products together.
For example, with vectors \( \langle 5, -0.4 \rangle \) and \( \langle 1.6, 20 \rangle \), the dot product is \( 5 \times 1.6 + (-0.4) \times 20 \).
This results in \( 8 + (-8) = 0 \), clearly a sum of zero, indicating that these vectors are orthogonal.

vector operations

Working with vectors involves several operations, and understanding these is crucial for making sense of vector mathematics. Basic vector operations include addition, subtraction, scalar multiplication, and the dot product. Each operation has its unique purpose and method.
Here is a brief overview:

• Addition: To add vectors, add their corresponding components. For instance, the sum of \( \langle x_1, y_1 \rangle \) and \( \langle x_2, y_2 \rangle \) is \( \langle x_1 + x_2, y_1 + y_2 \rangle \).
• Subtraction: Similar to addition, but for subtraction, you subtract the corresponding components of the vectors.
• Scalar Multiplication: Multiply each component of the vector by a scalar (a constant number). For example, multiplying \( \langle a, b \rangle \) by scalar \( c \) gives \( \langle ca, cb \rangle \).
• Dot Product: Previously mentioned, the dot product is an inner product that results in a scalar and helps determine orthogonality.

Vector operations form the building blocks of vector calculus and are often used in physics and engineering to solve various problems.

precalculus mathematics

Precalculus mathematics covers several essential topics that lay the groundwork for calculus and higher-level math courses. It includes the study of functions, trigonometry, and vectors. A strong foundation in precalculus concepts helps in understanding the behavior of mathematical systems and how they interact.
Here are key areas in precalculus:

• Functions: Learn about different types of functions, such as linear, quadratic, and polynomial functions, to model real-world situations.
• Trigonometry: Trigonometric functions, like sine and cosine, and their relationships are crucial, especially in geometry and calculus.
• Vectors: As discussed, vectors are a central topic; understanding them is important for geometry and physics applications.

Mastering these areas equips students with the skills to approach calculus with confidence and to apply mathematical concepts in practical scenarios.