Rapid Recall
- When we write the sum $\vec F = \vec F_{\parallel \vec d}+ \vec F_{\perp \vec d}$, we call $\vec F_{\parallel \vec d}$ the ____ of $\vec F$ that is ____ to $\vec d$.
- What do we call $\vec F_{\perp \vec d}$.
- If a force $\vec F$ of magnitude 10 N acts in the same direction as a displacement $\vec d$ of 5 m, what is the work done by $\vec F$ through the displacement $\vec d$?
Group problems
Let $\vec F=(-10,0)$ and $\vec d=(2,1)$. Recall the projection of $\vec F$ onto $\vec d$ is $\ds \text{proj}_\vec d\vec F = \frac{\vec F\cdot \vec d}{\vec d\cdot \vec d}\vec d$.
- Compute the projection of $\vec F$ onto $\vec d$ (so compute $\vec F_{\parallel \vec d}$).
- Draw $\vec F$, $\vec d$ and $\text{proj}_\vec d\vec F $ on the same grid, all with their base at the origin. Try your best to give the $x$ and $y$ directions the same scale, otherwise you won't be able to see the connections among vectors.
- Add to your picture the vector difference $\vec F_{\perp \vec d}=\vec F - \text{proj}_\vec d\vec F $. Which vectors in your picture are orthogonal?
- Draw $\text{proj}_\vec F\vec d $, without doing any computations. Make sure you compare with neighbors at some point.
- Draw the vector field $\vec F(x,y) = \langle2x+y,x+2y\rangle$. (Based at $(x,y)$, draw the vector $\langle2x+y,x+2y\rangle$.)
- Draw the vector field $\vec F(x,y) = \langle y,-x\rangle$.
- How much work is done by $\vec F$ through the displacement $\vec d$?
- Let $x=2u+3$ and $y=4v-5$. Complete the $u,v,x,y$ table below, and then construct a graph of both $u^2+v^2=1$ (in the $uv$-plane) and the corresponding equation in the $xy$-plane. $$ \begin{array}{c|c|c|c} u&v&x&y\\\hline 0&0&3&-5\\ 1&0&5&-5\\ 0&1&&\\ -1&0&&\\ 0&-1&&\\ \end{array} $$
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