Vector Representations

“Write down a representation of a 2D vector.”

This SWBQ is intended to refresh students' memories of the varying representations of vectors seen throughout undergraduate math and physics courses. Further, this SWBQ segues well into an introductory discussion of Bra-Ket notation.

Make sure that both the component representation $$ \vec{v} = x\hat{x} + y\hat{y}$$ and the column (or row) representation $$\vec{v} = \begin{pmatrix} x\\y \end{pmatrix}$$ are mentioned if no students offer them as a response.

This can be a great opportunity to introduce the “representation” operator $\dot{=}$ to indicate that two equated vectors are not equal, but that one is a representation of the other in a different basis. Students can then be introduced to equations such as $$\vec{v} \,\dot{=} \begin{pmatrix} v_x\\v_y \end{pmatrix}\, ,$$ which implies that the vector $\vec{v}$ components $v_x$ and $v_y$ in the $x$ and $y$ basis, respectively.

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