Problem 17 - Calculus for Biology and Medicine

Question

Find the dot product of \(\mathbf{x}=[0,-1,3]^{\prime}\) and \(\mathbf{y}=[-3,0,1]^{\prime}\).

Step-by-Step Solution

Answer

The dot product is 3.

Step 1: Understand the Dot Product

The dot product of two vectors \(\mathbf{x} = [x_1, x_2, x_3]^{\prime}\) and \(\mathbf{y} = [y_1, y_2, y_3]^{\prime}\) is calculated as \(x_1y_1 + x_2y_2 + x_3y_3\).

Step 2: Identify the Corresponding Elements

Identify the corresponding elements of the vectors: \(x_1 = 0\), \(x_2 = -1\), \(x_3 = 3\) from \(\mathbf{x}\) and \(y_1 = -3\), \(y_2 = 0\), \(y_3 = 1\) from \(\mathbf{y}\).

Step 3: Compute Each Product

Calculate the individual products: \(0 \cdot (-3) = 0\), \((-1) \cdot 0 = 0\), and \(3 \cdot 1 = 3\).

Step 4: Sum the Products

Sum the individual products from Step 3 to find the dot product: \(0 + 0 + 3 = 3\).

Key Concepts

vector mathematics

Vector mathematics is a fundamental area of mathematics that deals with quantities having both magnitude and direction. Vectors are represented in a coordinate system and can be used to describe physical phenomena such as displacement, velocity, and force in physics. There are several operations that can be performed on vectors to manipulate and combine them in different ways.

In the context of our exercise, we are interested in one specific operation: the dot product. This operation is crucial because it helps determine how much one vector aligns with another. The resultant value of a dot product can tell us about the angle between the vectors, as a zero dot product indicates they are perpendicular, while a positive or negative product indicates the angle is less than 90 degrees or more than 90 degrees, respectively.

Understanding vector mathematics is key to grasping more complex topics in physics and engineering disciplines.

vector calculations

In vector calculations, one often needs to find either the sum, difference, or product of vectors. One common form of multiplication involving vectors is the dot product, also known as the scalar product.

The dot product is defined as the sum of the products of the corresponding components of two vectors. For example, if we have vectors \(\mathbf{x} = [x_1, x_2, x_3]'\) and \(\mathbf{y} = [y_1, y_2, y_3]'\), the dot product is calculated as:

• \(x_1 \times y_1\)
• \(x_2 \times y_2\)
• \(x_3 \times y_3\)

Summing these products gives you the final scalar result.

In our given exercise, after computing each of these individual products, we end up with \(0 + 0 + 3 = 3\), which is the dot product of the vectors \(\mathbf{x}\) and \(\mathbf{y}\). Since dot product outputs a scalar, it reduces multi-dimensional vectors into a single number, simplifying complex calculations.

linear algebra

Linear algebra forms the backbone of vector mathematics and provides tools for vector calculations. It is the study of vectors, vector spaces, linear transformations, and systems of linear equations. A prominent feature of linear algebra is its ability to handle high-dimensional data efficiently and to provide solutions in many areas of mathematics and applied sciences.

When dealing with vectors in linear algebra, understanding operations such as the dot product is crucial. The dot product not only provides information about the relationship between two vectors, like their directional similarity but also aids in solving linear equations and optimizing problems found in various fields like computer graphics, machine learning, and optimization.

By mastering the basics of linear algebra, such as vector operations, one gains essential skills for advanced studies and practical applications in technology and science.