Vectors Explained: The Basics

Let's explore vectors, fundamental mathematical objects that combine magnitude and direction.Unlike scalar quantities that only have size, vectors tell us both how much and which way.Vectors are typically represented by arrows on a coordinate system. The length of the arrow shows the magnitude, while its orientation shows the direction.Here's a force vector, showing both strength and direction of the push or pull.And here's a velocity vector, indicating both speed and direction of motion.Finally, this displacement vector shows how far and in which direction an object has moved.Let's look at some common real-world examples where vectors are used.The magnitude of a vector represents its size or strength.Here are three vectors with different magnitudes but the same direction.Any vector can be broken down into its horizontal and vertical components.Here's a vector with coordinates two comma three.We can break this vector into its x and y components.The x-component has a magnitude of two units along the horizontal axis.And the y-component has a magnitude of three units in the vertical direction.There are several ways to write this vector using standard notation.We can write it as an ordered pair: two comma threeOr using unit vectors: two i plus three jOr as a column vector with two above threeLet's examine the components in more detail.The original vector is the sum of its x and y components.Let's explore how to add vectors using the tip-to-tail method.First, let's draw vector a in blue.Now we place vector b starting at the tip of vector a.The resultant vector, shown in green, goes directly from the start of a to the tip of b.Vector subtraction can be thought of as adding the negative of a vector.To subtract vector b, we first find negative b by reversing its direction.Then we add this negative vector to vector a using the same tip-to-tail method.Scalar multiplication changes a vector's magnitude but keeps its direction.Multiplying by 2 doubles the vector's length.Multiplying by one-half makes the vector half as long.And multiplying by negative one reverses the direction.Let's review the key points about vector operations.These fundamental operations form the basis for more advanced vector mathematics.