Problem 9 - Trigonometry

Question

Find each of the following dot products $(3,4) \cdot\langle 5,5\rangle

Step-by-Step Solution

Answer

The dot product is 35.

Step 1: Understand Dot Product Formula

The dot product of two vectors \((a, b)\) and \((c, d)\) is found using the formula: \[ a imes c + b imes d \] Here, vector \((3,4)\) corresponds to \((a, b)\) and vector \(\langle 5, 5 \rangle\) corresponds to \((c, d)\).

Step 2: Multiply Corresponding Components

Using the formula, multiply the corresponding components of both vectors: - First components: \(3 \times 5\) - Second components: \(4 \times 5\)

Step 3: Calculate Individual Products

Calculate the results of the products you set in Step 2:- Calculate \(3 \times 5 = 15\)- Calculate \(4 \times 5 = 20\)

Step 4: Add Products Together

Add the results from Step 3 together to find the dot product:- Add \(15 + 20 = 35\)

Key Concepts

Vector Multiplication

Vector multiplication is a fundamental operation in vector algebra. When we talk about multiplying vectors, we often refer to the dot product. The dot product results in a scalar, which is essentially a single number. To calculate the dot product of two vectors

• Label the components of the first vector as \(a, b\) and of the second vector as \(c, d\).
• Apply the formula for the dot product: \( a \times c + b \times d \).

This formula tells us to multiply each corresponding pair of components from both vectors and then sum these products together. It's important to remember that the dot product measures the extent to which two vectors point in the same direction.

Trigonometry

Trigonometry often finds its way into vector operations, especially with dot products. A remarkable aspect of the dot product is its connection to the angle between two vectors. Using trigonometry, we can express the dot product as:

• \( \text{Dot Product} = |\vec{A}| \cdot |\vec{B}| \cdot \cos(\theta) \), where \(\theta\) is the angle between the vectors.

This relationship shows that the dot product is not just about multiplication but also about understanding angles. If you're familiar with cosine, you'll recognize that if two vectors point in the exact same direction, \(\theta\) is zero degrees, and \(\cos(\theta)\) becomes 1, maximizing the dot product. Conversely, if the vectors are perpendicular, \(\cos(\theta)\) is 0, making the dot product zero.

Vector Components

Vectors are often broken down into components for easier manipulation. Each vector in a plane can be described by its horizontal and vertical components:

• These components are usually expressed as \(x\) and \(y\), forming an ordered pair \( (x, y) \).
• The concept here is that a vector can be thought of as a combination of its directional parts along these axes.

By using components, we simplify complex geometric and physical problems into more manageable numerical calculations. For example, in our original problem, the components \(3, 4\) and \(5, 5\) were utilized to perform straightforward arithmetic operations for the dot product. Understanding the importance of vector components is crucial for various applications in physics and engineering, where vectors represent forces, velocities, and other directional quantities.