Welcome to this lesson on vector formulas. Vectors are mathematical objects that have both magnitude and direction. In this series, we will explore the most important formulas related to vectors, including addition, subtraction, dot product, cross product, and magnitude calculations. Let's begin by visualizing two vectors in a coordinate system.The first important formula is vector addition. When we add two vectors, we add their corresponding components. Geometrically, we place the tail of the second vector at the head of the first vector. The sum is the vector from the origin to the final point. This is also known as the parallelogram rule or triangle rule.Vector subtraction is closely related to addition. When we subtract vector b from vector a, we subtract their corresponding components. Geometrically, the difference vector a minus b points from the head of vector b to the head of vector a. This can also be viewed as adding the negative of vector b to vector a.The dot product, also called scalar product, is a fundamental operation that produces a scalar value. It equals the sum of products of corresponding components, or equivalently, the product of magnitudes times the cosine of the angle between vectors. The dot product measures how much two vectors align with each other and is used to calculate projections.