In physics we break vectors into components that point along the coordinate axes.  These vector components are treated separately, then put back together.

A vector component, or just component, is a part of a vector that points along a coordinate axis.  We can break (or resolve) any vector in an x-y plane an x- component and a y-component.  Vectors in the x,y,z space are broken into three components.

Vector componentsThe diagram shows a vector, its angle, and its components.  We draw the components as dashed lines here, to make sure that we realize that only one vector is present. Now  notice that we could add the two components to make the original vector.  Likewise, we could use some trigonometry to determine the components from the vector itself.  (We’ll do both in the next few lessons.)

Why use Vector Components?

Chess move as vector componentsImagine we can move in either the x-direction or the y-direction, but not both at the same time.  This is similar to the limitation on the movement of the knight in the game of chess.  The player can move the piece two spaces in one direction, and then one space in the other dimension.  In the figure, we see one such allowed move.  Notice that we can write this particular move as \(\color{black}{2 \hat{y} + \hat{x}}\).  We could also write it as \(\color{black}{\hat{x} +2 \hat{y}}\), the preferred way in physics.

Vector Components in Kinematics

We have seen the kinematics equations in one dimension.  If we have velocities and/or accelerations in two dimensions, trying to develop a single set of equations mixing the x and y components would be quite complicated.  If we break any vector into x and y components, we can just use two sets of equations, one for each coordinate direction.  These equations would be…

\(\color{black}{x = x_o + v_{ox} \Delta t + \frac{1}{2} a_x (\Delta t)^2}\),

\(\color{black}{v_x = v_{ox} +  a_x \Delta t }\), and

\(\color{black}{v_x^2 = v_{ox}^2 + 2 a_x (x – x_o)}\) for the x-direction, and…

\(\color{black}{y = y_o + v_{oy} \Delta t + \frac{1}{2} a_y (\Delta t)^2}\),

\(\color{black}{v_y = v_{oy} +  a_y \Delta t }\), and

\(\color{black}{v_x^2 = v_{oy}^2 + 2 a_y (y – y_o)}\) for the y-direction.

We would treat each dimension separately, with the \(\color{black}{\Delta t}\) as a shared variable.