Problem 15 - 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 is 1.

Step 1: Understand the Dot Product Concept

The dot product of two vectors is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. For two vectors \( \mathbf{x} = [x_1, x_2]^{\prime} \) and \( \mathbf{y} = [y_1, y_2]^{\prime} \), the dot product is calculated as \( x_1 \times y_1 + x_2 \times y_2 \).

Step 2: Identify Components of the Vectors

Given vectors are \( \mathbf{x} = [1, 2]^{\prime} \) and \( \mathbf{y} = [3, -1]^{\prime} \). Identify their components: \( x_1 = 1, x_2 = 2, y_1 = 3, \) and \( y_2 = -1 \).

Step 3: Multiply Corresponding Components

Multiply each corresponding component of the vectors: \( x_1 \times y_1 = 1 \times 3 \) and \( x_2 \times y_2 = 2 \times (-1) \). This gives us \( 1 \times 3 = 3 \) and \( 2 \times (-1) = -2 \).

Step 4: Sum the Products

Add the products obtained from the previous step: \( 3 + (-2) = 1 \). This is the dot product of the two vectors.

Key Concepts

Vectors

Imagine vectors as arrows that have both a direction and a length. Vectors represent quantities that are defined by both their magnitude and direction.
For example, wind velocity is a vector: it blows in a specific direction and with a certain speed. Similarly, in a 2D plane, vectors like \([1,2]\) or \([3,-1]\) have specific coordinates that describe their direction and length in that plane.
Vectors can also be expressed in different forms:

• Graphically, as arrows on a coordinate system.
• Algebraically, using numbers that identify the vector’s coordinates.

These coordinates, or components, tell you how far to move along each axis to reach the vector’s endpoint from the origin.

Algebraic Operation

The dot product is a specific type of algebraic operation performed on vectors. It’s a way to multiply vectors to return a single scalar rather than a vector.
This operation involves multiplying corresponding components of two vectors and then summing these products. For instance, if you have vectors \(\mathbf{x} = [1, 2]\) and \(\mathbf{y} = [3, -1]\), the dot product involves:

• Multiplying the first components: \(1 \times 3 = 3\)
• Multiplying the second components: \(2 \times (-1) = -2\)

The sum of these results gives you the dot product: \(3 + (-2) = 1\).
Algebraically speaking, the dot product turns vector multiplication into a straightforward arithmetic task, simplifying calculations in both physics and engineering.

Coordinate Vectors

Coordinate vectors, such as \([1, 2]\) and \([3, -1]\), represent points or movements in space using numerical values. These numbers denote the vector’s position relative to the origin of a coordinate system.
When you break it down, each value in the coordinate vector shows the movement along a particular axis. For instance, in the vector \([1, 2]\):

• The first coordinate, 1, tells you the movement along the x-axis.
• The second coordinate, 2, tells you the movement along the y-axis.

It’s vital to pick out these components correctly, especially when performing operations like the dot product. Understanding the coordinate system and how vectors are situated within it helps grasp complex mathematical and physical concepts.