In this small group activity, students draw components of a vector in Cartesian and polar bases. Students then write the components of the vector in these bases as both dot products with unit vectors and as bra/kets with basis bras.
- In the first figure, draw the projections of \(\vec{v}\) onto the rectangular basis vectors \(\hat{x}\) and \(\hat{y}\).
- In the second figure, draw the projections of \(\vec{v}\) onto the polar basis vectors \(\hat{s}\) and \(\hat{\phi}\).
- Write the components of \(\vec{v}\) as dot products: \begin{align*} v_x &= \hskip 100pt& v_s&=\\ \\ v_y &= & v_{\phi}&= \end{align*}
- Write the components of \(\vec{v}\) as bra/kets: (Warning: Often, particularly in quantum mechanics settings, we will assume that all kets are states and therefore normalized so that their probability is one. We are NOT making that assumption here. Write the vector \(\vec{v}=\left|{v}\right\rangle \) without worrying about its normalization.) \begin{align*} v_x &= \hskip 100pt& v_s&=\\ \\ v_y &= & v_{\phi}&= \end{align*}
Students work in small groups to write the components of an arbitrary vector in both dot product notation and bra/ket notation.