Problem 9 - Precalculus with Limits

Question

In Exercises 7-14, find the dot product of \(\mathbf{u}\) and \(\mathbf{v}\). \(\mathbf{u} = \langle -4, 1 \rangle\) \(\mathbf{v} = \langle 2, -3 \rangle\)

Step-by-Step Solution

Answer

The dot product of vectors \(\mathbf{u}\) and \(\mathbf{v}\) is \(-11\).

Step 1: Identify the components of the vectors

Identify the components of each vector for substitution into the dot product formula. For vector \(\mathbf{u}\), the components are \(-4\) and \(1\). For vector \(\mathbf{v}\), the components are \(2\) and \(-3\).

Step 2: Apply the dot product formula

Calculate the dot product by substituting the components into the formula as follows: \(\mathbf{u} \cdot \mathbf{v} = (-4) * 2 + 1 * (-3)\).

Step 3: Simplify the expression

Simplify the resulting expression from the last step by evaluating \(-4 * 2\) and \(1 * -3\): \(-8 + (-3) = -11\).

Key Concepts

Vector Components

Vectors are essential in representing quantities that have both magnitude and direction. Understanding their components is fundamental for many mathematical and physical contexts. A vector in a two-dimensional space can be represented by an ordered pair of numbers, often noted within angle brackets like \(\langle a, b \rangle\). These numbers are the **components of the vector**.

For the given vectors in the exercise:

• Vector \( \mathbf{u} = \langle -4, 1 \rangle \), has components \(-4\) and \(1\).
• Vector \( \mathbf{v} = \langle 2, -3 \rangle \), has components \(2\) and \(-3\).

Each component is crucial as it defines the vector's direction on each axis. The first number \(a\) refers to the horizontal direction (x-axis), and the second number \(b\) pertains to the vertical direction (y-axis). Understanding these components allows us to perform various operations and manipulations such as calculating the dot product effectively.

Vector Operations

Vector operations are mathematical processes that allow for the manipulation and combination of vectors. One important operation is the **dot product**. The dot product of two vectors \( \mathbf{u} = \langle a_1, b_1 \rangle \) and \( \mathbf{v} = \langle a_2, b_2 \rangle \) is calculated using the following formula:

\[\mathbf{u} \cdot \mathbf{v} = a_1 \times a_2 + b_1 \times b_2\]

Applying this formula involves multiplying the corresponding components of the vectors and summing the results. This provides information about the angle between the vectors and the magnitude of their interaction in directional space.

In the given exercise:

• The first vector \( \mathbf{u} \) has components \(-4\) and \(1\).
• The second vector \( \mathbf{v} \) has components \(2\) and \(-3\).

Thus, the dot product is computed as \((-4 \times 2) + (1 \times -3)\). The result \(-11\) shows how these vectors correlate to each other in the space they occupy.

Mathematics Problem-Solving

Mathematics problem-solving skills are essential when tackling exercises like calculating the dot product. Follow a structured approach to solve problems efficiently.

Start by identifying the task clearly; in this case, we want to find the dot product. Next, determine the required information, which involves recognizing the vector components and applying the correct formula.

This involves:

• Identifying components of \( \mathbf{u} \) and \( \mathbf{v} \).
• Applying the dot product formula accurately.
• Simplifying the arithmetic expressions to reach a final answer.

In the exercise, this structured process leads to understanding the result \(-11\). Developing these problem-solving techniques not only aids in mathematics but also enhances critical thinking and analytical skills across various disciplines.