Problem 11 - Trigonometry

Question

Find each of the following dot products \((-23,4) \cdot(15,-6)\)

Step-by-Step Solution

Answer

The dot product is -369.

Step 1: Understanding the Dot Product

The dot product of two vectors \((a, b)\) and \((c, d)\) in 2D is calculated using the formula: \((a \cdot c) + (b \cdot d)\). We'll apply this formula to our vectors.

Step 2: Identify the Components

For the vectors \((-23, 4)\) and \(15, -6)\), we have \(a = -23\), \(b = 4\), \(c = 15\), and \(d = -6\).

Step 3: Calculate the Products

Compute the individual products: \(-23 \cdot 15\) and \(4 \cdot (-6)\). Calculating these gives us \(-345\) and \(-24\), respectively.

Step 4: Add the Results

Add the products from the previous step to find the dot product: \(-345 + (-24)\). This equals \(-345 - 24 = -369\).

Key Concepts

Vectors

Vectors are essential in mathematics, especially when dealing with physics and engineering problems. In a simple sense, a vector is a quantity that possesses both magnitude and direction. Vectors can vary in dimensions, commonly being 2D, 3D, or even higher.
In a two-dimensional (2D) space, vectors are described as ordered pairs \(a, b\), where \(a\) represents the component along the x-axis and \(b\) represents the component along the y-axis. For instance, the vector \((-23, 4)\) shows a movement of \(-23\) units in the x-direction and \(4\) units in the y-direction.
Vectors depictions simplify complex problems by illustrating effects like direction and force visually, aiding in understanding the essence of interactions in varied disciplines.

2D Vector Operations

Manipulating vectors involves operations that help perform different tasks connected to directions and magnitudes. One common operation is the dot product, also known as the scalar product.
Other operations include:

• Addition: The sum of two vectors such as \(a_1, b_1\) and \(a_2, b_2\) results in a new vector \(a_1 + a_2, b_1 + b_2\).
• Subtraction: The difference between two vectors is \(a_1 - a_2, b_1 - b_2\).
• Scalar multiplication: Multiplying a vector \(a, b\) by a number \k\ results in \(ka, kb\), changing its magnitude.

Understanding these basics equips you with the tools to address more intricate calculations involving two-dimensional vectors.

Dot Product Formula

The dot product is a useful operation when determining the relative orientation of two vectors. It produces a scalar (a single number) rather than another vector.
The formula for calculating the dot product in 2D is \(a \cdot c + b \cdot d\), where \(a\) and \(b\) are components of the first vector, and \(c\) and \(d\) are components of the second vector. For example, the dot product of \((-23, 4)\) and \(15, -6)\) is computed as:

• Multiply \(-23 \cdot 15 = -345\).


• Multiply \(4 \cdot (-6) = -24\).


• Add the results to get the dot product: \(-345 - 24 = -369\).

The dot product can tell you if two vectors are perpendicular: if the dot product is zero, the vectors are orthogonal.

Vector Components

Each vector consists of components. In 2D, these are often denoted as the x-component and the y-component. These components are essentially projections along the coordinate axes.
The components allow you to deconstruct the vector into simpler parts. For example, the x-component of \((-23, 4)\) is \(-23\), showing its action along the x-axis, while \(4\) is the y-component that shows its action along the y-axis.
Having a deep understanding of vector components helps in performing vector operations and in visualizing scenarios, such as displacement or force, within two dimensional spaces. Recognizing and utilizing vector components empowers the analysis and resolution of various mechanical problems across different fields.