Linear Combinations and Alternative Coordinate Systems

How to Read a Coordinate System

Let’s take a fresh look at the familiar rectangular coordinate system. In the diagram on the left, the point \((3,2)\) can be located by moving your finger 3 units along the \(x\)-axis, then moving your finger 2 units parallel to the \(y\)-axis. We can think of the \(x\) and \(y\) axes as being determined by the standard unit vectors \(\vec {i}\) and \(\vec {j}\). The vector \(\begin{bmatrix}3\\2\end{bmatrix}\) can be written as a linear combination of \(\vec {i}\) and \(\vec {j}\) as

\[\begin{bmatrix}3\\2\end{bmatrix}=3\vec {i}+2\vec {j}\]

[Picture] [Picture]

Orthogonal unit vectors are often a convenient choice for establishing a coordinate system, but occasionally, using other vectors is preferrable. An example of an alternative coordinate grid determined by vectors \(\vec {v}_1\) and \(\vec {v}_2\) is shown below.

[Picture]

Fortunately, finding points and constructing vectors in such coordinate systems relies on the same principles as we use in the familiear rectangular coordinate system. To reach point \(P\), we first travel 2 units along the line containing \(\vec {v}_1\), then one unit along the line parallel to \(\vec {v}_2\). We can say that in this coordinate system, \(P\) has coordinates \((2,1)\). Also, observe that vector \(\vec {x}\) can be written as a linear combination of \(\vec {v}_1\) and \(\vec {v}_2\) as follows

\[\vec {x}=2\vec {v}_1+\vec {v}_2\]

The following interactive demonstrates how to find the coordinates for a point using a coordinate grid defined using vectors \(\overrightarrow {OA}\) and \(\overrightarrow {OB}\) (in that order). To use the interactive, you can either (1) move the sliders to change the coordinates of \(P\) within this coordinate system, or (2) move the tips of vectors \(\overrightarrow {OA}\) and \(\overrightarrow {OB}\) to change the coordinate system. For (2), explain why the coordinates of \(P\) do not change while \(P\) moves around the plane.

Explorations

Use the interactive below to answer the questions

List the coordinates for each \(P_i\) with respect to the given coordinate system. Your coordinates should be of the form \((A\text {-coordinate}, B\text {-coordinate})\).
\[P_1=\left (\answer {1},\answer {1}\right )\]
\[P_2=\left (\answer {-2},\answer {0}\right )\]
\[P_3=\left (\answer {0},\answer {-1}\right )\]
\[P_4=\left (\answer {-2},\answer {3}\right )\]
\[P_5=\left (\answer {3},\answer {-2}\right )\]
Move point \(B\) to coincide with \(P_3\) List the coordinates for each \(P_i\) with respect to the new coordinate system. Your coordinates should be of the form \((A\text {-coordinate}, B\text {-coordinate})\).
\[P_1=\left (\answer {1},\answer {-1}\right )\]
\[P_2=\left (\answer {-2},\answer {0}\right )\]
\[P_3=\left (\answer {0},\answer {1}\right )\]
\[P_4=\left (\answer {-2},\answer {-3}\right )\]
\[P_5=\left (\answer {3},\answer {2}\right )\]
How do these coordinates compare to the coordinates in the previous question? Explain why this is happening. (Hint: you can use the RESET button to return to the original coordinate system to compare.)
Press RESET to return to the original coordinate system. Move point \(A\) to coincide with \(P_1\) List the coordinates for each \(P_i\) with respect to the new coordinate system. Your coordinates should be of the form \((A\text {-coordinate}, B\text {-coordinate})\).
\[P_1=\left (\answer {1},\answer {0}\right )\]
\[P_2=\left (\answer {-2},\answer {2}\right )\]
\[P_3=\left (\answer {0},\answer {-1}\right )\]
\[P_4=\left (\answer {-2},\answer {5}\right )\]
\[P_5=\left (\answer {3},\answer {-5}\right )\]
How do these coordinates compare to the coordinates in the original coordinate system? Explain why this is happening. (Hint: you can use the reset button to return to the original coordinate system to compare.)
Press RESET to return to the original coordinate system. Express each \(\overrightarrow {OP}_i\) as a linear combination of \(\overrightarrow {OA}\) and \(\overrightarrow {OB}\).
\[\overrightarrow {OP}_1=\answer {1}\overrightarrow {OA}+\answer {1}\overrightarrow {OB}\]
\[\overrightarrow {OP}_2=\answer {-2}\overrightarrow {OA}+\answer {0}\overrightarrow {OB}\]
\[\overrightarrow {OP}_3=\answer {0}\overrightarrow {OA}+\answer {-1}\overrightarrow {OB}\]
\[\overrightarrow {OP}_4=\answer {-2}\overrightarrow {OA}+\answer {3}\overrightarrow {OB}\]
\[\overrightarrow {OP}_5=\answer {3}\overrightarrow {OA}+\answer {-2}\overrightarrow {OB}\]

Discuss the relationship between your answers to the first question and your answers here.

Move point \(B\) to coincide with \(P_2\). What do you observe? Do vectors \(\overrightarrow {OA}\) and \(\overrightarrow {OB}\) determine a good coordinate system for the plane? Can we express every point in the plane as a linear combination of \(\overrightarrow {OA}\) and \(\overrightarrow {OB}\)? Can we express some points in the plane as linear combinations of \(\overrightarrow {OA}\) and \(\overrightarrow {OB}\)?
We will use the same set-up as in the previous exploration but introduce an additional vector \(\overrightarrow {OC}\).

Suppose we want to express \(P_1\) using a coordinate system determined by three vectors \(\overrightarrow {OA}\), \(\overrightarrow {OB}\), and \(\overrightarrow {OC}\). If the coordinates are to be of the form \((A\text {-coordinate}, B\text {-coordinate}, C\text {-coordinate})\), how many ways do you think there would be to express \(P_1\)? Fill in the missing coordinates for \(P_1\) below.
\[P_1=\left (\answer {1},\answer {1},0\right )\]
\[P_1=\left (0,\answer {2},\answer {0.5}\right )\]
\[P_1=\left (\answer {2}, 0, \answer {-0.5}\right )\]
\[P_1=\left (\answer {3},-1,-1\right )\]
Based on your work above, express \(\overrightarrow {OP_1}\) as a linear combination of \(\overrightarrow {OA}\), \(\overrightarrow {OB}\), and \(\overrightarrow {OC}\). How many ways do you think there are to do this?
Compare and contrast the coordinate systems in this exploration and Exploration .

Existence and Uniqueness of Representation