Problem 62 - Precalculus

Question

Find the dot product \langle 11,12\rangle\(\cdot\langle-2,3\rangle\) Solution: Multiply the outer and inner components. $$ \langle 11,12\rangle \cdot\langle-2,3\rangle=(11)(3)+(12)(-2) $$ Simplify. \(\langle 11,12\rangle \cdot\langle-2,3\rangle=9\) This is incorrect. What mistake was made?

Step-by-Step Solution

Answer

The mistake was in the multiplication of components. Correct multiplication gives a dot product of 14.

Step 1: Understand the Dot Product Formula

The dot product of two vectors \( \langle a, b \rangle \) and \( \langle c, d \rangle \) is calculated using the formula: \( a \cdot c + b \cdot d \). This means you multiply the corresponding components from each vector and then sum the results.

Step 2: Identify Components of Vectors

Identify the components of each vector. For the vector \( \langle 11, 12 \rangle \), \( a = 11 \) and \( b = 12 \). For the vector \( \langle -2, 3 \rangle \), \( c = -2 \) and \( d = 3 \).

Step 3: Multiply Corresponding Components

Using the formula, calculate \( a \cdot c = 11 \cdot (-2) = -22 \) and \( b \cdot d = 12 \cdot 3 = 36 \).

Step 4: Sum the Results

Now, add the results of the multiplications: \( -22 + 36 \).

Step 5: Simplify the Expression

Calculate the sum: \( -22 + 36 = 14 \). Thus, the dot product is 14, not 9.

Key Concepts

Vectors

Vectors are a fundamental concept in mathematics and physics. They are entities that possess both a magnitude and a direction. This makes them incredibly useful for describing physical quantities like force, velocity, or displacement. In mathematics, vectors are often represented by arrows, where the length of the arrow corresponds to the vector's magnitude, and the direction of the arrow indicates the direction of the vector itself.

For instance, in a two-dimensional plane, a vector can be written as \( \langle x, y \rangle \), where \( x \) and \( y \) are the components of the vector. These components determine how far and in what direction you move along the x-axis and y-axis, respectively. Further, vectors can be added together, subtracted, and multiplied by scalars, making them versatile tools in various calculations. Understanding vectors is key to mastering many advanced topics in mathematics and science.

Vector Components

Vector components are the building blocks of vectors. When we describe a vector using its components, we refer to how much the vector stretches along each axis. For example, in the vector \( \langle 11, 12 \rangle \), 11 is the component along the x-axis, and 12 is the component along the y-axis.

Components are essential because they break down a vector into simpler parts that can be easily manipulated. This decomposition is what allows for the calculation of vector operations like the dot product. By working with individual components, we simplify complex vector problems, transforming them into more manageable arithmetic operations.

• Magnitude: The component values help calculate the vector's magnitude or length using the Pythagorean theorem.
• Direction: The ratio of the components determines the vector's direction in space.

By understanding and manipulating these components, you gain better control over vector calculations, which is crucial in many scientific and engineering applications.

Vector Operations

Vector operations encompass a variety of manipulations that can be performed on vectors, such as addition, scalar multiplication, and more complex operations like the dot product. Each type of operation serves a different purpose, allowing you to analyze and solve diverse mathematical and physical problems.

Dot Product:
The dot product, one of the most common vector operations, involves multiplying two vectors to obtain a scalar. For vectors \( \langle a, b \rangle \) and \( \langle c, d \rangle \), the dot product is calculated as \( a \cdot c + b \cdot d \). This operation is fundamental because it provides information about the angle between the vectors, specifically determining the degree of orthogonality; a dot product of zero indicates perpendicular vectors.

Vector operations extend beyond just arithmetic algorithms. They play a crucial role in computer graphics, physics simulations, and machine learning. Mastering these operations allows for efficient problem-solving and deeper insights into how vectors describe and predict the behavior of real-world systems.