Understanding the magnitude of a vector is fundamental in the study of vectors in physics and mathematics. The magnitude of a vector describes its length and is a scalar quantity. In this section, we will explore how to calculate the magnitude of a vector given its components along the x and y axes.

Introduction to Vector Magnitude

Vectors are mathematical entities with both magnitude and direction. The magnitude of a vector (often denoted as |v|) is a measure of its length. For a two-dimensional vector v with components x and y, the magnitude can be calculated using the Pythagorean theorem.

Calculating Magnitude

The magnitude of a vector v = $(x, y)$ is given by the formula:

This formula arises from the Pythagorean theorem, applied to a right-angled triangle formed by the components of the vector.

Image courtesy of Cue Math

Key Concepts

• Vector Components: The x and y values that define the direction and magnitude of the vector in two dimensions.
• Pythagorean Theorem: A fundamental relation in Euclidean geometry among the three sides of a right triangle.
• Square Root: The function that returns the positive square root of a number.

Example 1: Basic Calculation

Given vector v = $(3, 4)$, calculate its magnitude.

Magnitude = 5 units

Example 2: Negative Components

Given vector v = $(-6, 8)$, calculate its magnitude.

Magnitude = 10 units

Example 3: Decimal Components

Given vector v = $(0.5, -2.5)$, calculate its magnitude.

Magnitude ≈ 2.55 units

Practice Questions

Question 1:

For vector $(7, -24)$, calculate its magnitude.

Magnitude = 25 units

Question 2:

For vector $(-3.5, 2.5)$, calculate its magnitude.

Magnitude ≈ 4.3 units

Tips for Success

• Remember to square the components before adding them; this eliminates any negative signs.
• The magnitude is always a positive value or zero.
• Practice with vectors in different quadrants to become comfortable with positive and negative components.