Problem 13 - Precalculus with Limits

Question

In Exercises 7-14, find the dot product of \(\mathbf{u}\) and \(\mathbf{v}\). \(\mathbf{u} = 3\mathbf{i} + 2\mathbf{j}\) \(\mathbf{v} = -2\mathbf{i} - 3\mathbf{j}\)

Step-by-Step Solution

Answer

The dot product of \(\mathbf{u}\) and \(\mathbf{v}\) is -12.

Step 1: Identify the Components of the Vectors

The components of the vectors \(\mathbf{u}\) and \(\mathbf{v}\) are identified. For \(\mathbf{u}\), the components are 3 and 2. For \(\mathbf{v}\), the components are -2 and -3.

Step 2: Multiply Corresponding Components

The corresponding components of the vectors are multiplied. For the \(\mathbf{i}\) (or x) components, \(3 \cdot -2 = -6\). For the \(\mathbf{j}\) (or y) components, \(2 \cdot -3 = -6\).

Step 3: Sum the Results

The results of the multiplications are summed to find the dot product. The dot product \(\mathbf{u} \cdot \mathbf{v}\) is thus \(-6 + -6 = -12\).

Key Concepts

Vector Components

Understanding the concept of vector components is crucial when dealing with vectors in mathematics and physics. A vector is an entity that has both a magnitude and a direction, often represented in a 2-dimensional plane with components along the x and y axes.

When you break down a vector like \( \mathbf{u} = 3\mathbf{i} + 2\mathbf{j} \), you're identifying its components:

• x-component: This is the \( 3\mathbf{i} \), indicating a movement or influence in the horizontal direction.
• y-component: This is the \( 2\mathbf{j} \), indicating a movement or influence in the vertical direction.

These components are essential for calculations such as the dot product, as they allow us to assess how vectors interact in each dimensional plane individually.

In the given exercise, the vector \( \mathbf{v} = -2\mathbf{i} - 3\mathbf{j} \) consists of:

• x-component: \(-2\mathbf{i}\)
• y-component: \(-3\mathbf{j}\)

By breaking down vectors into their components, performing calculations like vector multiplication becomes structured and manageable.

Vector Multiplication

Vector multiplication, specifically the dot product, is a fundamental way of combining two vectors to produce a scalar value. This operation is useful in various mathematical and physical contexts, such as calculating how much one vector influences another.

When performing the dot product, you simply multiply corresponding components of the vectors and then add these products together. Here's how it works with the example vectors \( \mathbf{u} \) and \( \mathbf{v} \):

• For \( \mathbf{i} \) (x-components): Multiply \( 3 \times (-2) = -6 \)
• For \( \mathbf{j} \) (y-components): Multiply \( 2 \times (-3) = -6 \)

This step-by-step multiplication reveals the interaction between corresponding components, reflecting how one vector projects onto another. The negative result highlights the opposite directionality between the vectors' components. Once you multiply and sum all the corresponding components, you're left with the dot product of the vectors.

Vector Addition

Vector addition involves combining two or more vectors so that their magnitudes and directions result in a singular vector often called the resultant vector.While vector addition is not directly required for calculating the dot product, understanding it enhances comprehension of how different vectors combine in space.

To perform vector addition, simply add each pair of corresponding components:

• Add the x-components: For vectors \(\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j}\) and \(\mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j}\), it is \(a_1 + b_1\)
• Add the y-components: Similarly, add \(a_2 + b_2\)

The resultant vector will combine these sums, yielding \((a_1 + b_1)\mathbf{i} + (a_2 + b_2)\mathbf{j}\).

Vector addition becomes intuitive once you realize that each vector acts as a directional arrow in space. By bringing them together, they point to a new location or direction.This concept is analogous to combining forces or directions, providing a holistic view when analyzing vectors and their interactions in multidimensional spaces. By mastering vector addition, you enhance your capability in breaking down complex vector problems efficiently.