Problem 18 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Dot Product

The dot product of two vectors \( \mathbf{a} = [a_1, a_2, a_3]^{\prime} \) and \( \mathbf{b} = [b_1, b_2, b_3]^{\prime} \) is given by the formula \( a_1b_1 + a_2b_2 + a_3b_3 \). It produces a scalar (a single number).

Step 2: Identify Components of Vectors

Identify the components of vectors \( \mathbf{x} \) and \( \mathbf{v} \). For \( \mathbf{x} = [2, -3, 1]^{\prime} \), we have components \( x_1 = 2 \), \( x_2 = -3 \), and \( x_3 = 1 \). For \( \mathbf{v} = [3, 1, -2]^{\prime} \), we have components \( v_1 = 3 \), \( v_2 = 1 \), and \( v_3 = -2 \).

Step 3: Calculate Each Product of Components

Multiply the corresponding components of the vectors: \( 2 \times 3 = 6 \), \( -3 \times 1 = -3 \), and \( 1 \times -2 = -2 \).

Step 4: Sum the Results

Add the results of the products from Step 3: \( 6 + (-3) + (-2) = 6 - 3 - 2 \).

Step 5: Final Calculation

Compute \( 6 - 3 - 2 \) to find the dot product. This results in \( 1 \).

Key Concepts

Vector Calculus

Vector calculus is a branch of mathematics that deals with vector-related concepts and operations. In this context, vectors are mathematical entities that have both magnitude and direction. They are incredibly useful for representing physical quantities like force, velocity, and acceleration. In vector calculus, operations like addition, subtraction, and dot product are integral.
The dot product is one such operation and is fundamental to vector calculus. It results in a scalar, simplifying complex multidimensional calculations. It helps in finding angles between vectors, determining vector projections, and calculating work done by force fields. Understanding these concepts enriches your problem-solving toolkit in physics and engineering.

Scalar Multiplication

Scalar multiplication is a basic operation in vector calculus that involves multiplying a vector by a scalar, or a single number. This operation scales the vector, changing its magnitude without affecting its direction.
Imagine you have a vector representing a force, and you scale this vector by multiplying it with a scalar. It changes the force's magnitude, making it stronger or weaker, without altering its initial direction.

• Scalar multiplication retains the direction of the vector.
• It modifies the magnitude by the scalar's absolute value.
• Negative scalar flips the vector's direction.

In our dot product example, scalar multiplication is vital because it computes the product of corresponding vector components before adding them up for the final dot product calculation.

Vector Components

Vector components are individual parts that make up the whole vector. Each component represents the vector's projection along an axis in Cartesian coordinates. For example, a three-dimensional vector \( \mathbf{x} = [x_1, x_2, x_3]^{\prime}\) has three components: \( x_1, x_2, \) and \( x_3\). These are important because they allow us to perform vector operations like the dot product.
Identifying vector components was a key step in solving the original exercise. For vectors \( \mathbf{x} = [2, -3, 1]^{\prime} \) and \( \mathbf{v} = [3, 1, -2]^{\prime}\), their components guided the scalar multiplications and highlighted the importance of individual axis contributions in vector calculations.
By mastering vector components, you enhance your comprehension of vector behavior in different dimensions, laying a solid foundation for more advanced studies and applications.