Vectors and their magnitudes are geometric quantities, independent of coordinates and choice of basis
In the small town of Coriander, the library can be found by starting at the center of the town square, walking 25 meters north (\(\boldsymbol{\vec{a}}\)), turning \(90^\circ\)
to the right, and walking a further 60 meters (\(\boldsymbol{\vec{b}}\)).
Draw a figure showing the displacement vectors \(\boldsymbol{\vec{a}}\) and \(\boldsymbol{\vec{b}}\), as well as
their sum, the displacement vector \(\boldsymbol{\vec{v}}=\boldsymbol{\vec{a}}+\boldsymbol{\vec{b}}\).
How far is the library from the center of the town square?
Let \(\boldsymbol{\hat{x}}\) be the unit vector pointing east, and \(\boldsymbol{\hat{y}}\) be the unit vector pointing north. Express \(\boldsymbol{\vec{a}}\), \(\boldsymbol{\vec{b}}\), and \(\boldsymbol{\vec{v}}\) in terms of \(\boldsymbol{\hat{x}}\) and
\(\boldsymbol{\hat{y}}\).
It turns out that magnetic north in Coriander is approximately \(14^\circ\) degrees east of true north. If you use a compass to find the library (!), the above directions will fail. Instead, you must walk 39 meters in the direction of magnetic north (\(\boldsymbol{\vec{A}}\)), turn \(90^\circ\) to the right, and walk a further 52
meters (\(\boldsymbol{\vec{B}}\)).
Draw a figure showing the displacement vectors \(\boldsymbol{\vec{A}}\) and \(\boldsymbol{\vec{B}}\), as well as
their sum, the displacement vector \(\boldsymbol{\vec{v}}=\boldsymbol{\vec{A}}+\boldsymbol{\vec{B}}\).
How far is the library from the center of the town square?
Let \(\boldsymbol{\hat{X}}\) be the unit vector pointing towards “magnetic east”, and \(\boldsymbol{\hat{Y}}\) be the unit vector pointing towards magnetic north. Express \(\boldsymbol{\vec{A}}\), \(\boldsymbol{\vec{B}}\), and
\(\boldsymbol{\vec{v}}\) in terms of \(\boldsymbol{\hat{X}}\) and \(\boldsymbol{\hat{Y}}\).
Can any vector displacement within the town limits be expressed as the sum of two vectors, one of which points north and the other east?