Problem 1 - Calculus for Biology and Medicine

Question

Let \(\mathbf{x}=[1,4,-1]^{\prime}\) and \(\mathbf{y}=[-2,1,0]^{\prime}\). (a) Find \(\mathbf{x}+\mathbf{y}\). (b) Find \(2 \mathbf{x}\). (c) Find \(-3 \mathbf{y}\).

Step-by-Step Solution

Answer

(a) \([-1, 5, -1]\). (b) \([2, 8, -2]\). (c) \([6, -3, 0]\).

Step 1: Adding Two Vectors

To find \( \mathbf{x} + \mathbf{y} \), we add the corresponding components of the vectors \( \mathbf{x} \) and \( \mathbf{y} \). The components of \( \mathbf{x} \) are \( [1, 4, -1] \) and the components of \( \mathbf{y} \) are \( [-2, 1, 0] \).So, \([1 + (-2), 4 + 1, -1 + 0] = [-1, 5, -1]\).

Step 2: Scaling a Vector by a Constant

To find \( 2 \mathbf{x} \), multiply each component of \( \mathbf{x} = [1, 4, -1] \) by 2.So, \( [2 \times 1, 2 \times 4, 2 \times (-1)] = [2, 8, -2] \).

Step 3: Scaling Another Vector by a Constant

To find \( -3 \mathbf{y} \), multiply each component of \( \mathbf{y} = [-2, 1, 0] \) by -3.So, \( [-3 \times (-2), -3 \times 1, -3 \times 0] = [6, -3, 0] \).

Key Concepts

Vector Scaling

Vector scaling is a process where each component of a vector is multiplied by a constant (known as a scalar). This operation changes the magnitude (size) of the vector without altering its direction, unless the scalar is negative, which would additionally reverse the vector's direction.

Here's a simple example: Consider a vector \( \mathbf{v} = [2, 3] \). If we scale this vector by a scalar of 3, then each component is multiplied by 3 resulting in a new vector \( 3\mathbf{v} = [3 \times 2, 3 \times 3] = [6, 9] \). The vector now is three times longer but still points in the same direction.

When scalars are negative, as in the case of finding \(-3 \mathbf{y}\) from our exercise, not only is the length of the vector scaled by three times, but its direction also flips. That's why \( \mathbf{y} = [-2, 1, 0] \) turns into \( [-3 \times (-2), -3 \times 1, -3 \times 0] = [6, -3, 0] \), lengthening the vector and reversing its original direction.

Vector Components

Each vector can be imagined as a collection of scalar values along coordinates. These values are called vector components. They define the vector’s size along each of its dimensions or axes.

For example, in a three-dimensional space, a vector \( \mathbf{x} = [1, 4, -1] \) means:

• The vector moves 1 unit along the x-axis.
• It stretches 4 units along the y-axis.
• It moves -1 unit along the z-axis.

Each dimension's movement is independent, and this breakdown into components helps in various vector operations, such as addition and scaling. For the vector addition in our problem, where \( \mathbf{x} + \mathbf{y} = [-1, 5, -1] \), each component of \( \mathbf{x} \) is added to the corresponding component of \( \mathbf{y} \), showing how components interact in calculations.

Linear Algebra

Linear Algebra is a branch of mathematics that deals with vector spaces and linear mappings between these spaces. It's the foundational part of dealing with vectors and matrices, which are represented as arrays of numbers.

Key concepts include:

• Vectors: Lists of numbers that could represent points in space.
• Vector Operations: Include addition, scaling, and dot product.
• Matrices: Rectangular arrays of numbers representing complex transformations and systems of linear equations.

The tasks of vector addition and scaling in our exercise fall under linear algebra's umbrella. Linear algebra provides the rules for manipulating these vectors, enabling them to tackle varied and complex mathematical problems.

Linear algebra is widely used in different fields like physics, engineering, computer science, and more because it offers powerful methods to model and solve real-world problems.