3.4 - Vector components and rectangular notation
Rectangular notation: Defining a vector by its
components.
Often what we know, or want to know, about a particular vector is not its overall
magnitude and direction, but how far it extends horizontally and vertically. On a graph,
we represent the horizontal direction as x and the vertical direction as y. These are
called Cartesian coordinates. The xcomponent of a vector indicates its extent in the
horizontal dimension and the ycomponent its extent in the vertical dimension.
Rectangular notation is a way to describe a vector using the components that make up
the vector. In rectangular notation, the x and y components of a vector are written
inside parentheses. A vector that extends a units along the x axis and b units along the
y axis is written as (a,b). For instance (3, 4) is a vector that extends positive three in
the x direction and positive four in the y direction from its starting point.
The components of vectors are scalars with the direction indicated by their sign: x
components point right (positive) or left (negative), and y components point up (positive)
or down (negative). You see the x and y components of a car’s velocity vector in
Concept 1 at the right, shown as “hollow” vectors. The x and y values define the vector,
as they provide direction and magnitude.
For a vector A, the x and y components are sometimes written as Ax and Ay. You see
this notation used for a velocity vector v in Equation 1 and Example 1 on the right.
Consider the car shown in Example 1 on the right. Its velocity has an x component vx of
17 m/s and a y component vy of í13 m/s. We can write the car’s velocity vector as
(17, í13) m/s.
A vector can extend in more than two dimensions: z represents the third dimension.
Sometimesz is used to represent distance toward or away from you. For instance, your
computer monitor’s width is measured in the x dimension, its height with y and your
distance from the monitor with z. If you are reading this on a computer monitor and
punch your computer screen, your fist would be moving in the z dimension. (We hope
we’re not the cause of any such aggressive feelings.) Three-dimensional vectors are
written as (x,y,z).
The z component can also represent altitude. A Tour de France bike racer might believe
the z dimension to be the most important as he ascends one of the competition’s
famous climbs of a mountain pass.
Vector components and
rectangular notation
x component and y component