Problem 15 - Calculus for Biology and Medicine

Question

Find the dot product of \(\mathbf{x}=[-1,2]^{\prime}\) and \(\mathbf{y}=[-3,-4]^{\prime}\).

Step-by-Step Solution

Answer

The dot product of \(\mathbf{x}\) and \(\mathbf{y}\) is \(-5\).

Step 1: Understand the dot product

The dot product of two vectors \( \mathbf{x} = [x_1, x_2]' \) and \( \mathbf{y} = [y_1, y_2]' \) is calculated as the sum of the products of their corresponding components: \( x_1 \cdot y_1 + x_2 \cdot y_2 \).

Step 2: Identify the components of the vectors

For vectors \( \mathbf{x} = [-1, 2]' \) and \( \mathbf{y} = [-3, -4]' \), the components are as follows: \( x_1 = -1 \), \( x_2 = 2 \), \( y_1 = -3 \), and \( y_2 = -4 \).

Step 3: Calculate the products of corresponding components

Calculate each product: \( x_1 \cdot y_1 = (-1) \cdot (-3) = 3 \) and \( x_2 \cdot y_2 = 2 \cdot (-4) = -8 \).

Step 4: Sum up the products

Add the results from the previous step: \( 3 + (-8) = -5 \). Thus, the dot product of \( \mathbf{x} \) and \( \mathbf{y} \) is \(-5\).

Key Concepts

Vector Multiplication

Vector multiplication involves combining two vectors to extract meaningful information about their relationship. One common form is the dot product, also known as the scalar product. The dot product offers insight into the angle between vectors and their magnitudes. It is calculated through the multiplication of corresponding components and summing the results.

For example, with vectors \( \mathbf{x} = [-1,2]' \) and \( \mathbf{y} = [-3,-4]' \), the dot product is computed as \( x_1 \cdot y_1 + x_2 \cdot y_2 \).
This approach, where each component of one vector multiplies its corresponding component in the other, demonstrates the core principle of vector multiplication.

Remember:

• The dot product of two perpendicular vectors is always zero.
• The dot product of two identical vectors results in the square of its magnitude.

Understanding this can help solve broader problems in physics and engineering, where vector relationships are crucial.

Linear Algebra

Linear algebra is a branch of mathematics concerned with vector spaces and linear mappings between these spaces. It encompasses a variety of operations, including vector multiplication. Dot products, for instance, are fundamental operations within linear algebra.

These operations allow for solutions to complex systems of equations, transformations of geometric data, and the simplification of mathematical modeling in various fields.
Working with vectors through linear algebra applications helps in understanding multi-dimensional spaces and interactions.

Within this framework:

• Operations like the dot product are used to examine vector magnitude and direction.
• Linear combinations of vectors form the basis for much of computational mathematics.

By mastering vector operations, students gain a powerful toolset for exploring mathematical concepts.

Vector Components

Vector components are the building blocks of a vector in a given space. They describe the vector’s position based on axes of a coordinate system. For instance, the vector \( \mathbf{x} = [-1,2]' \) has components \( x_1 = -1 \) and \( x_2 = 2 \).

These components are critical as they allow for understanding the vector's behavior in relation to others, facilitating operations like the dot product. By multiplying and summing these components, we derive meaningful metrics about the vectors such as directionality and magnitude comparison.

Key points about vector components include:

• They define the vector’s reach along each axis in its coordinate system.
• In operations such as the dot product, components reveal how two vectors align or oppose.

Grasping the role of vector components paves the way for advanced applications in spatial analysis and physics.