10.2 Standard Vectors
We have two standard vectors in the \( (x, y) \)-coordinate plane:
-
the standard \( x \)-vector, \( \vec{\standard}_{x} : \Vector_{(x, y)} \), travels one East, and
-
the standard \( y \)-vector, \( \vec{\standard}_{y} : \Vector_{(x, y)} \), travels one North.
As before, we'll use cardinal directions to discuss movement in the \( (x, y) \)-coordinate plane.
Standard vectors give us a uniform way to work with vectors. We may write any vector as a combination of the standard vectors.
Definition.
Any vector \( v : \Vector_{(x, y)} \) can be written in standard form as
\[
\begin{align}
v &= a \mult \vec{\standard}_{x} + b \mult \vec{\standard}_{y}
\end{align}
\]
where the numbers
\[
\begin{align}
a &: \Number
&
&\andSpaced
&
b &: \Number
\end{align}
\]
are called the parts of \( v \).
Let's think through how standard form works, geometrically. Suppose \( v : \Vector_{(x, y)} \) is a vector that travels some amount \( a : \Number \) to the East and some amount \( b : \Number \) to the North.
The vector \( a \mult \vec{\standard}_{x} \) travels \( a \) to the East, and the vector \( b \mult \vec{\standard}_{y} \) travels \( b \) to the North. Adding these together gives our vector \( v \).
This same process works for the other cardinal directions, too! The vectors
\[
\begin{align}
&- \vec{\standard}_{x}
&
&- \vec{\standard}_{y}
\end{align}
\]
travel one West and one South, respectively. So we'll use a negative \( x \)-part \( a \) to represent traveling West, and we'll use a negative \( y \)-part \( b \) to represent traveling South.
Example.
Consider the vector \( v : \Vector_{(x, y)} \) as pictured.
This vector travels three to the West and one North.
\[
\begin{align}
v &= -3 \mult \vec{\standard}_{x} + \vec{\standard}_{y}
\end{align}
\]
Our vector has \( x \)-part negative three and \( y \)-part one.
The Zero Vector
The zero vector
\[
\begin{align}
0 &: \Vector_{(x, y)}
\\
0 &= 0 \mult \vec{\standard}_{x} + 0 \mult \vec{\standard}_{y}
\end{align}
\]
represents a lack of movement. As such, the zero vector is difficult to draw as an arrow: it has no length nor direction. But it is a useful object for vector arithmetic. Subtracting a vector from itself gives the zero vector.
\[
\begin{align}
v - v &= 0
\end{align}
\]
And adding the zero vector to any vector has no effect.
\[
\begin{align}
v + 0 &= v
\end{align}
\]
We use the same symbol for the number zero and the vector zero,
\[
\begin{align}
0 &: \Number
&
&\andSpaced
&
0 &: \Vector_{},
\end{align}
\]
but it's best to think of these as distinct objects.