A vector is a fundamental mathematical concept that represents quantities with both magnitude and direction. Unlike scalars which only have size, vectors contain directional information. For example, velocity is a vector because it tells us both how fast something is moving and in which direction.
Every vector can be decomposed into components along the coordinate axes. The x-component shows how far the vector extends horizontally, while the y-component shows the vertical extension. These components form a right triangle with the original vector as the hypotenuse.
Vector addition follows the head-to-tail rule. We place the tail of the second vector at the head of the first vector. The resultant vector goes from the tail of the first vector to the head of the second vector. This can also be visualized using the parallelogram method.
To find the magnitude of a vector, we use the Pythagorean theorem: the magnitude equals the square root of the sum of the squares of its components. The direction angle is found using the arctangent of the y-component divided by the x-component.