Problem 11 - Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The dot product of \(\mathbf{u}\) and \(\mathbf{v}\) is 6.

Step 1: Identify the Vectors

In this exercise, we have two vectors: \(\mathbf{u} = 4\mathbf{i} - 2\mathbf{j}\) and \(\mathbf{v} = \mathbf{i} - \mathbf{j}\). These vectors are provided in component form, \(\mathbf{i}\) being the x-component and \(\mathbf{j}\) being the y-component.

Step 2: Compute the Dot Product

The dot product between two vectors is calculated by multiplying corresponding components of the vectors, and then taking the sum of those products. For our vectors, it will become: \( \mathbf{u} \cdot \mathbf{v} = (4)(1) + (-2)(-1) \).

Step 3: Evaluate the Expression

Now we simply have to compute the expression, which will become: \( \mathbf{u} \cdot \mathbf{v} = 4+2=6\).

Key Concepts

Vector Components

Understanding vector components is the key to mastering vector algebra. A vector in two-dimensional space can be expressed using its components along the x-axis and y-axis. This is often represented using unit vectors, namely \(\mathbf{i}\) and \(\mathbf{j}\).

For instance, the vector \(\mathbf{u} = 4\mathbf{i} - 2\mathbf{j}\) has components of 4 in the x-direction and -2 in the y-direction. Similarly, \(\mathbf{v} = \mathbf{i} - \mathbf{j}\) is made up of components 1 and -1 in the x and y directions respectively.

• The x-component represents the horizontal direction.
• The y-component represents the vertical direction.
• Unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) help to simplify writing.

Understanding these components allows us to further manipulate vectors through operations like addition, subtraction, and the dot product.

Vector Operations

Vector operations are essential in physics, engineering, and mathematics, providing the groundwork for manipulating vectors. One of the key operations is the dot product, which combines two vectors into a scalar, reflecting how much one vector extends in the direction of another.

The dot product, especially, is a vital operation and can be calculated as follows for any vectors \(\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j}\) and \(\mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j}\):

• Multiply each corresponding pair of components: \(a_1 \cdot b_1\) and \(a_2 \cdot b_2\).
• Sum these products: \(a_1b_1 + a_2b_2\).

For vectors \(\mathbf{u} = 4\mathbf{i} - 2\mathbf{j}\) and \(\mathbf{v} = \mathbf{i} - \mathbf{j}\), the dot product is computed as \( (4)(1) + (-2)(-1) = 6 \).
This scalar value is key in determining concepts such as orthogonality and projection, forming a basis for more complex vector analysis.

Precalculus Exercises

Precalculus exercises, such as the dot product calculation, provide foundational skills for future calculus learning. Such exercises help students grasp the integration of algebra and geometry through practical applications of vectors.

By performing these exercises, students build an understanding of:

• Using vector notation to represent physical quantities.
• Applying mathematical operations to solve real-world problems.
• Visualizing concepts geometrically in space.

The dot product in \(\mathbf{u}\) and \(\mathbf{v}\), as calculated, looks simple yet assists in tackling more sophisticated mathematical and physical problems later. Mastery of these exercises paves the way to understanding limits, derivatives, and integrals by laying the groundwork in vector calculus.