Vectors | Chapter 1, Essence of linear algebra

Introduction

This tutorial introduces the essential concepts of vectors, as discussed in the first chapter of the linear algebra series by 3Blue1Brown. Understanding vectors is crucial for grasping more complex topics in mathematics and physics, as they represent quantities that have both magnitude and direction.

Step 1: Understanding Vectors

• Vectors are mathematical objects that have both a direction and a magnitude.
• They can be represented in a coordinate system, typically in two dimensions, as [x, y].
• Each vector can be thought of as an arrow pointing from the origin (0,0) to the point (x, y).

Practical Advice

• To visualize vectors, draw arrows on a grid. The length represents the magnitude, and the arrow points in the direction of the vector.

Step 2: Vector Addition

• The sum of two vectors can be found by adding their corresponding components.

• For two vectors A = [x1, y1] and B = [x2, y2], the result of A + B is given by:

  A + B = [x1 + x2, y1 + y2]
  

Practical Advice

• When adding vectors graphically, you can use the tip-to-tail method. Place the tail of the second vector at the tip of the first vector, and the resultant vector goes from the tail of the first vector to the tip of the second.

Step 3: Scalar Multiplication

• A vector can be multiplied by a scalar (a single number), which alters its magnitude but not its direction.

• For a vector A = [x, y] and a scalar k, the product is:

  kA = [kx, ky]
  

Common Pitfalls to Avoid

• Remember that multiplying by a negative scalar will reverse the direction of the vector.

Step 4: Real-World Applications of Vectors

• Vectors are used in various fields, including physics to represent forces, velocities, and accelerations.
• In computer graphics, vectors define the position of objects in space.

Conclusion

In this tutorial, we've covered the basics of vectors, including their definition, vector addition, and scalar multiplication. Understanding these concepts lays the groundwork for more advanced topics in linear algebra. As a next step, consider exploring how vectors interact in three dimensions and their applications in real-world scenarios, such as physics and engineering.