Problem 9 - Calculus for Biology and Medicine

Question

Find the length of \(\mathbf{x}=[0,1,5]^{\prime}\).

Step-by-Step Solution

Answer

The length of the vector \(\mathbf{x} = [0, 1, 5]^\prime\) is \(\sqrt{26}\).

Step 1: Understanding the Length of a Vector

The length of a vector, also known as the vector's magnitude or norm, is a measure of the size of the vector. For a vector \(\mathbf{x} = [x_1, x_2, x_3]^\prime\), the length is calculated using the formula \(\|\mathbf{x}\| = \sqrt{x_1^2 + x_2^2 + x_3^2}\). In this exercise, \(\mathbf{x} = [0, 1, 5]^\prime\).

Step 2: Substituting Vector Components into the Formula

Using the formula for the length of a vector, substitute the components of \(\mathbf{x}\) into the formula: \(\|\mathbf{x}\| = \sqrt{0^2 + 1^2 + 5^2}\).

Step 3: Calculate the Squares of Each Component

Calculate the square of each component: \(0^2 = 0\), \(1^2 = 1\), and \(5^2 = 25\).

Step 4: Add the Squared Components

Add the squared components together: \(0 + 1 + 25 = 26\).

Step 5: Take the Square Root of the Sum

Find the square root of the sum: \(\|\mathbf{x}\| = \sqrt{26}\).

Step 6: Finalize the Answer

The length of the vector \(\mathbf{x}\) is \(\sqrt{26}\).

Key Concepts

Vector Norm

The vector norm, often referred to simply as the norm, is an essential concept in vector mathematics. It gives us a sense of how long or large a vector is. When you find the norm of a vector, you're essentially finding its length in the geometric space.
The most common vector norm is the Euclidean norm, which is calculated as:

• Given a vector \(\mathbf{x} = [x_1, x_2, x_3 ]^\prime \), the formula is \(\|\mathbf{x}\| = \sqrt{x_1^2 + x_2^2 + x_3^2} \).

Norms are fundamental because they help measure distances and sizes of objects represented as vectors. There are other types of norms, such as the "1-norm" or "infinity norm," but the Euclidean norm is the most prevalent in basic vector calculations.

Vector Length

The vector length is another term often interchangeable with vector magnitude or vector norm. It's a measure of the straight-line distance from the initial point of the vector to its terminal point. This idea is intuitive as it reflects how long the vector "stretches" across its space.
To calculate the vector length, you essentially use the same Euclidean formula applied in the norm:

• \(\|\mathbf{x}\| = \sqrt{x_1^2 + x_2^2 + x_3^2} \)

When you have a vector, knowing its length can help in a variety of mathematical contexts, such as physics, where vectors often represent things like velocity or force.

Euclidean Distance

The concept of Euclidean distance stems from Euclidean geometry, which is the mathematical study of shapes, space, and their relationships. The Euclidean distance between two points in space is the length of a straight line connecting them. When considering a vector, it describes the distance from the vector's endpoint to the origin point.
For a vector originating from the origin, the length of the vector itself is also its Euclidean distance. The formula you use for this is identical to the vector norm and length:

• \(\|\mathbf{x}\| = \sqrt{x_1^2 + x_2^2 + x_3^2} \)

The Euclidean distance is crucial in many fields, including computer science and physics, to compare distances between multidimensional points or positions.

Vector Components

Vector components are the building blocks of a vector. Each component points in a different direction of the vector's space, commonly corresponding to the x, y, and z dimensions. In our original exercise, the vector \([0, 1, 5]^\prime\) has components 0, 1, and 5, representing its movement along those respective axes.
Understanding each component helps reveal how the vector behaves in a Cartesian coordinate system:

• The vector \([0, 1, 5]^\prime\) means no movement along x, a unit movement along y, and five units along z.

Components are pivotal in calculations, as you substitute them into formulas for magnitude, direction, and even dot products. Each component plays its part when finding the vector norm or Euclidean distance, highlighting how vectors stretch and orient themselves in space.