Problem 16 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The dot product of \( \mathbf{x} \) and \( \mathbf{y} \) is 1.

Step 1: Understanding the dot product

The dot product (or scalar product) of two vectors \( \mathbf{x} \) and \( \mathbf{y} \) is calculated by multiplying corresponding components of the vectors and then summing the results. For vectors \( \mathbf{x} = [x_1, x_2] \) and \( \mathbf{y} = [y_1, y_2] \), the dot product is given by \( x_1y_1 + x_2y_2 \).

Step 2: Component-wise Multiplication

Multiply the corresponding components of \( \mathbf{x} \) and \( \mathbf{y} \):1. Multiply the first components: \( 1 \times 3 = 3 \).2. Multiply the second components: \( 2 \times (-1) = -2 \).

Step 3: Summing the Products

Add the products obtained from the component-wise multiplication: \( 3 + (-2) = 1 \).

Step 4: Final Result

The dot product of the vectors \( \mathbf{x} = [1, 2]' \) and \( \mathbf{y} = [3, -1]' \) is 1.

Key Concepts

Vector Operations

Vector operations are key tools in mathematics and physics that allow us to manipulate vectors to obtain useful information. Vectors are entities that have both magnitude and direction, and are often represented in coordinate form such as \(\mathbf{x} = [1, 2]'\). The most basic vector operations include addition, subtraction, and scalar multiplication.
When performing vector addition, you add the corresponding components of two vectors to create a new vector. Similarly, subtraction involves taking the difference between corresponding components. Scalar multiplication entails multiplying each component of a vector by a scalar (a single number), adjusting the magnitude of the vector.
Understanding vector operations is fundamental, as they form the basis of more advanced techniques and computations in areas like physics, engineering, and computer graphics.

Scalar Product

The scalar product, often referred to as the dot product, is a fundamental concept in linear algebra and vector calculus. The scalar product offers a way to multiply two vectors, resulting in a scalar rather than another vector. This operation can help in determining the angle between two vectors or in assessing their alignment.
To calculate the scalar product for two vectors, such as \(\mathbf{x} = [x_1, x_2]\) and \(\mathbf{y} = [y_1, y_2]\), you multiply each pair of corresponding components and then sum these products:

• Multiply \(x_1\) and \(y_1\).
• Multiply \(x_2\) and \(y_2\).
• Add the results of these multiplications to get a single number.

For example, the dot product of \(\mathbf{x} = [1, 2]\) and \(\mathbf{y} = [3, -1]\) is calculated by \(1 \times 3 + 2 \times (-1) = 3 - 2 = 1\), resulting in a scalar value of 1.
The scalar product is extensively used in physics to define work done by a force and projection of vectors.

Component-wise Multiplication

Component-wise multiplication is a simple yet powerful operation when working with vectors. It involves multiplying the respective components of two vectors. In the context of the dot product, this step is crucial for obtaining meaningful results.
By multiplying vector components individually, you calculate intermediate products that eventually contribute to the final scalar, or dot product. Here’s how it works:

• Align the vectors vertically.
• Multiply each corresponding pair of components to produce a list of products.
• Sum these products to achieve the final result.

For vectors \([1, 2]\) and \([3, -1]\), component-wise multiplication gives us \(1 \times 3\) and \(2 \times (-1)\), yielding intermediate values 3 and -2, respectively. Adding these gives \(1\), which is the dot product.
This operation is straightforward and aids in many complex calculations, where understanding each component's contribution is vital.