Rapid Recall
- Give a vector equation of the line that passes through $P=(3,4)$ and $Q=(-2,5)$.
- Compute the dot product of $\vec P$ and $\vec Q$. $\quad$ Bonus: find the angle between $\vec P$ and $\vec Q$
- Give two vectors that are orthogonal to $\vec P$.
Group problems
Let $P=(3,4)$ and $Q=(-2,5)$.
- The vector projection of $\vec P$ onto $\vec Q$ is $\ds \text{proj}_\vec Q\vec P = \frac{\vec P\cdot \vec Q}{\vec Q\cdot \vec Q}\vec Q$. Compute the projection of $\vec P$ onto $\vec Q$.
- Draw $\vec P$, $\vec Q$ and $\text{proj}_\vec Q\vec P $ 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.
- Compute the projection of $\vec Q$ onto $\vec P$.
- Then draw $\vec P$, $\vec Q$ and $\text{proj}_\vec P\vec Q $ on the same grid, all with their base at the origin. Then add to your picture the vector difference $\vec Q - \text{proj}_\vec P\vec Q $.
- Draw two random vectors on your chalk board, with their base at the same point. Label them $\vec u$ and $\vec v$. Then draw both $\text{proj}_\vec v\vec u $ and $\text{proj}_\vec u\vec v $.
- How much work is done by $\vec P$ through a displacement $\vec Q$?
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