Introduction to Vectors

Vectors provide a concise mathematical framework for describing motion in three-dimensional space. For a moving ball, quantities such as displacement and velocity are naturally represented as vectors because they include both magnitude and direction. Geometrically, a vector is visualized as an arrow extending from one point to another. The length of the arrow corresponds to the vector’s magnitude, while its spatial orientation shows direction. This representation makes vectors especially useful for analyzing motion through the air, where movement occurs simultaneously along multiple axes.

A vector in three-dimensional space is commonly expressed in terms of its components along the x-, y-, and z-axes. These components are written in angle brackets, such as 〈x, y, z〉, and specify the extent of the vector in each coordinate direction. For example, the vector from the origin to the point (3, 2, 1) is written as 〈3, 2, 1〉. This means the displacement is 3 units in the x-direction, 2 units in the y-direction, and 1 unit in the z-direction. Because this vector begins at the origin, it is called a position vector.

When a vector connects two distinct points, A(x₁, y₁, z₁) and B(x₂, y₂, z₂), its components describe the net change in position from A to B. The vector is found by subtracting the coordinates of A from those of B:

\begin{equation*}\vec{AB} = \langle x_2 - x_1,; y_2 - y_1,; z_2 - z_1 \rangle\end{equation*}

These component differences show the displacement along each axis and are fundamental in tracking motion from one location to another.

The magnitude of a vector is its total length. In three dimensions, it is calculated using the Pythagorean Theorem in extended form:

\begin{equation*}|\mathbf{v}| = \bm{\sqrt{x^2 + y^2 + z^2}}\end{equation*}

This quantity gives the overall size of the displacement, independent of direction.