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.
![Vector Magnitude](https://tutorchase-resources-platform-client-production.vercel.app/_next/image?url=https%3A%2F%2Ftutorchase-production.s3.eu-west-2.amazonaws.com%2F5d1a9486-d5fe-4e3d-8259-c381a43dd640-file.png&w=1080&q=75)
Calculating Magnitude
The magnitude of a vector v = 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.
![Magnitude of a Vector Formula](https://tutorchase-resources-platform-client-production.vercel.app/_next/image?url=https%3A%2F%2Ftutorchase-production.s3.eu-west-2.amazonaws.com%2F901a48ce-ea4a-4083-b472-4a03e0f18de9-file.png&w=1080&q=75)
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 = , calculate its magnitude.
Magnitude = 5 units
![Magnitude of Vector (3,4)](https://tutorchase-resources-platform-client-production.vercel.app/_next/image?url=https%3A%2F%2Ftutorchase-production.s3.eu-west-2.amazonaws.com%2Fc6d7afc9-e36d-473f-b8d6-785dea05f3e7-file.png&w=1200&q=75)
Example 2: Negative Components
Given vector v = , calculate its magnitude.
Magnitude = 10 units
![Magnitude of Vector (-6,8)](https://tutorchase-resources-platform-client-production.vercel.app/_next/image?url=https%3A%2F%2Ftutorchase-production.s3.eu-west-2.amazonaws.com%2Fc993bc36-b270-4970-a3fe-c729a30223c6-file.png&w=1200&q=75)
Example 3: Decimal Components
Given vector v = , calculate its magnitude.
Magnitude ≈ 2.55 units
![Magnitude of Vector (0.5, -2.5)](https://tutorchase-resources-platform-client-production.vercel.app/_next/image?url=https%3A%2F%2Ftutorchase-production.s3.eu-west-2.amazonaws.com%2Ff033c49e-bb48-4a42-84e6-8d0e2fe1ca70-file.png&w=1200&q=75)
Practice Questions
Question 1:
For vector , calculate its magnitude.
Magnitude = 25 units
![Magnitude of Vector (7,-24)](https://tutorchase-resources-platform-client-production.vercel.app/_next/image?url=https%3A%2F%2Ftutorchase-production.s3.eu-west-2.amazonaws.com%2F0981acc0-a1dc-4172-ad81-d173bffe5983-file.png&w=1200&q=75)
Question 2:
For vector , calculate its magnitude.
Magnitude ≈ 4.3 units
![Magnitude of Vector (-3.5, 2.5)](https://tutorchase-resources-platform-client-production.vercel.app/_next/image?url=https%3A%2F%2Ftutorchase-production.s3.eu-west-2.amazonaws.com%2F975eb819-cf18-43a1-bec0-c2f54a96979e-file.png&w=1200&q=75)
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.