11.1 Standard Rulers

Before we give a general definition of rulers, let's spend a little time working with the standard rulers. Our goal for this section is to learn to make measurements using standard rulers. We'll see how measurement extends to all rulers in the next section.

We have two standard rulers for the \( (x, y) \)-coordinate plane: a standard \( x \)-ruler and a standard \( y \)-ruler.

\[ \begin{align} \bar{\standard}_{x} &: \Ruler_{(x, y)} & &\andSpaced & \bar{\standard}_{y} &: \Ruler_{(x, y)} \end{align} \]

The measurement product of a ruler \( r : \Ruler_{} \) and a vector \( v : \Vector_{} \) is a number

\[ \begin{align} \langle{r}\mathbin{|}{v}\rangle &: \Number. \end{align} \]

We write our ruler on the left side of a measurement and our vector on the right side.

Laws.

We enforce vector measurement laws

\[ \begin{align} &\langle{r}\mathbin{|}{v_1 + v_2}\rangle = \langle{r}\mathbin{|}{v_1}\rangle + \langle{r}\mathbin{|}{v_2}\rangle \\ &\langle{r}\mathbin{|}{n \mult v}\rangle = n \mult \langle{r}\mathbin{|}{v}\rangle \end{align} \]

where \( r : \Ruler_{} \) is a ruler, \( n : \Number \) is a number, and

\[ \begin{align} v_1 &: \Vector_{}, & v_2 &: \Vector_{}, & v &: \Vector_{} \end{align} \]

are vectors.

We also have standard measurements.

\[ \begin{align} \bigl\langle{\bar{\standard}_{x}}\mathbin{\big|}{\vec{\standard}_{x}}\bigr\rangle &= 1 & \bigl\langle{\bar{\standard}_{x}}\mathbin{\big|}{\vec{\standard}_{y}}\bigr\rangle &= 0 & \bigl\langle{\bar{\standard}_{y}}\mathbin{\big|}{\vec{\standard}_{x}}\bigr\rangle &= 0 & \bigl\langle{\bar{\standard}_{y}}\mathbin{\big|}{\vec{\standard}_{y}}\bigr\rangle &= 1 \end{align} \]

That's enough to us get started measuring with standard rulers!

Example.

Let's measure the vector

\[ \begin{align} v &: \Vector_{(x, y)} \\ v &= 2 \mult \vec{\standard}_{x} - 3 \mult \vec{\standard}_{y} \end{align} \]

with both of the standard rulers. We'll start by measuring with the standard \( x \)-ruler.

\[ \begin{align} \bigl\langle{\bar{\standard}_{x}}\mathbin{\big|}{v}\bigr\rangle &= \bigl\langle{\bar{\standard}_{x}}\mathbin{\big|}{2 \mult \vec{\standard}_{x} - 3 \mult \vec{\standard}_{y}}\bigr\rangle \\ &= \bigl\langle{\bar{\standard}_{x}}\mathbin{\big|}{2 \mult \vec{\standard}_{x}}\bigr\rangle - \bigl\langle{\bar{\standard}_{x}}\mathbin{\big|}{3 \mult \vec{\standard}_{y}}\bigr\rangle \\ &= 2 \mult \bigl\langle{\bar{\standard}_{x}}\mathbin{\big|}{\vec{\standard}_{x}}\bigr\rangle - 3 \mult \bigl\langle{\bar{\standard}_{x}}\mathbin{\big|}{\vec{\standard}_{y}}\bigr\rangle \\ &= 2 \mult 1 - 3 \mult 0 \\ &= 2 \end{align} \]

And now let's measure using the standard \( y \)-ruler.

\[ \begin{align} \bigl\langle{\bar{\standard}_{y}}\mathbin{\big|}{v}\bigr\rangle &= \bigl\langle{\bar{\standard}_{y}}\mathbin{\big|}{2 \mult \vec{\standard}_{x} - 3 \mult \vec{\standard}_{y}}\bigr\rangle \\ &= \bigl\langle{\bar{\standard}_{y}}\mathbin{\big|}{2 \mult \vec{\standard}_{x}}\bigr\rangle - \bigl\langle{\bar{\standard}_{y}}\mathbin{\big|}{3 \mult \vec{\standard}_{y}}\bigr\rangle \\ &= 2 \mult \bigl\langle{\bar{\standard}_{y}}\mathbin{\big|}{\vec{\standard}_{x}}\bigr\rangle - 3 \mult \bigl\langle{\bar{\standard}_{y}}\mathbin{\big|}{\vec{\standard}_{y}}\bigr\rangle \\ &= 2 \mult 0 - 3 \mult 1 \\ &= -3 \end{align} \]

In the previous example, the measurement \( \bigl\langle{\bar{\standard}_{x}}\mathbin{\big|}{v}\bigr\rangle \) is equal to the vector's \( x \)-part, and the measurement \( \bigl\langle{\bar{\standard}_{y}}\mathbin{\big|}{v}\bigr\rangle \) is equal to the vector's \( y \)-part. The same thing will happen for other vectors, too!

Formulas.

We can solve for the parts of a vector \( v : \Vector_{(x, y)} \) by measuring the vector with standard rulers. If

\[ \begin{align} v &= a \mult \vec{\standard}_{x} + b \mult \vec{\standard}_{y} \end{align} \]

then

\[ \begin{align} \bigl\langle{\bar{\standard}_{x}}\mathbin{\big|}{v}\bigr\rangle &= a & &\andSpaced & \bigl\langle{\bar{\standard}_{y}}\mathbin{\big|}{v}\bigr\rangle &= b. \end{align} \]

Why?

Let's check these formulas by measuring.

\[ \begin{align} \bigl\langle{\bar{\standard}_{x}}\mathbin{\big|}{v}\bigr\rangle &= \bigl\langle{\bar{\standard}_{x}}\mathbin{\big|}{a \mult \vec{\standard}_{x} + b \mult \vec{\standard}_{y}}\bigr\rangle & \bigl\langle{\bar{\standard}_{y}}\mathbin{\big|}{v}\bigr\rangle &= \bigl\langle{\bar{\standard}_{y}}\mathbin{\big|}{a \mult \vec{\standard}_{x} + b \mult \vec{\standard}_{y}}\bigr\rangle \\ &= \bigl\langle{\bar{\standard}_{x}}\mathbin{\big|}{a \mult \vec{\standard}_{x}}\bigr\rangle + \bigl\langle{\bar{\standard}_{x}}\mathbin{\big|}{b \mult \vec{\standard}_{y}}\bigr\rangle & &= \bigl\langle{\bar{\standard}_{y}}\mathbin{\big|}{a \mult \vec{\standard}_{x}}\bigr\rangle + \bigl\langle{\bar{\standard}_{y}}\mathbin{\big|}{b \mult \vec{\standard}_{y}}\bigr\rangle \\ &= a \mult \bigl\langle{\bar{\standard}_{x}}\mathbin{\big|}{\vec{\standard}_{x}}\bigr\rangle + b \mult \bigl\langle{\bar{\standard}_{x}}\mathbin{\big|}{\vec{\standard}_{y}}\bigr\rangle & &= a \mult \bigl\langle{\bar{\standard}_{y}}\mathbin{\big|}{\vec{\standard}_{x}}\bigr\rangle + b \mult \bigl\langle{\bar{\standard}_{y}}\mathbin{\big|}{\vec{\standard}_{y}}\bigr\rangle \\ &= a \mult 1 + b \mult 0 & &= a \mult 0 + b \mult 1 \\ &= a & &= b \end{align} \]