Problem 10 - Precalculus with Limits

Question

In Exercises \(7-14,\) find \(\mathbf{u} \cdot \mathbf{v}\) $$\mathbf{u}=\langle- 2,5\rangle$$ $$\mathbf{v}=\langle- 1,-8\rangle$$

Step-by-Step Solution

Answer

The dot product of vectors \(\mathbf{u}\) and \(\mathbf{v}\) is -38.

Step 1: Identify the components of the vectors

The components of vector \(\mathbf{u}\) are -2 and 5, and the components of vector \(\mathbf{v}\) are -1 and -8.

Step 2: Multiply the corresponding components

Now, multiply the corresponding components of the vectors: (-2) * (-1) and 5 * (-8). This will give 2 and -40 respectively.

Step 3: Add the products

Now add the products obtained in step 2. Hence, \( 2 + (-40) = -38 \)

Key Concepts

Understanding Vector Components

Vectors are essential in mathematics and physics. They help represent direction and magnitude. A vector in two dimensions has two components: an x-component and a y-component. Each component represents a part of the vector along the x-axis and y-axis, respectively.
In our example, vector \( \mathbf{u} = \langle -2, 5 \rangle \) has components -2 and 5.
This means:

• The x-component of \( \mathbf{u} \) is -2, showing movement to the left.
• The y-component of \( \mathbf{u} \) is 5, indicating movement upwards.

For vector \( \mathbf{v} = \langle -1, -8 \rangle \), the components -1 and -8 mean it moves left and down.
Understanding these components is crucial for performing operations like the dot product."},{

Performing Vector Operations: Dot Product

Vector operations are mathematical procedures performed on vectors.
The dot product is a common operation. It combines two vectors into a single scalar value. This result gives an indication of how aligned the vectors are. For two vectors \( \mathbf{u} = \langle a_1, b_1 \rangle \) and \( \mathbf{v} = \langle a_2, b_2 \rangle \), the dot product is calculated as:

• Multiply the x-components: \( a_1 \times a_2 \)
• Multiply the y-components: \( b_1 \times b_2 \)
• Add the results: \( (a_1 \times a_2) + (b_1 \times b_2) \)

In the example, \( \mathbf{u} = \langle -2, 5 \rangle \) and \( \mathbf{v} = \langle -1, -8 \rangle \):

• \((-2) \times (-1) = 2\)
• \(5 \times (-8) = -40\)

Adding them gives \(2 + (-40) = -38\).
This dot product shows how vectors relate in terms of direction and length.

Mathematical Calculations for Dot Products

Calculating the dot product involves simple arithmetic, yet it's a powerful tool. Here's how it's done step-by-step using vectors \( \mathbf{u} = \langle -2, 5 \rangle \) and \( \mathbf{v} = \langle -1, -8 \rangle \):
Firstly, identify the vector components. Here, \( \mathbf{u} \) has components -2 and 5; \( \mathbf{v} \) has -1 and -8.
Next, multiply corresponding components:

• \((-2) \cdot (-1) = 2\)
• \(5 \cdot (-8) = -40\)

Finally, add these products together:

• \(2 + (-40) = -38\)

This final result, -38, is the scalar "dot product" of the vectors.
This operation can help you determine angles and projections between vectors.