Let $P=(3,4)$ and $Q=(-1,2)$.
- Give a vector equation of the line that passes through $P$ and $Q$.
- Find the angle between $\vec P$ and $\vec Q$.
- Give two vectors that are orthogonal to $\vec P$.
- The 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 $\vec Q$. Then draw $\vec P$, $\vec Q$ and $\text{proj}_\vec Q\vec P $ on the same grid, all with their base at the origin.
- 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.
- 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 $.
- Draw the vector field $\vec F(x,y) = (2x+y,x+2y)$.
- Draw the vector field $\vec F(x,y) = \langle y,-x\rangle$.
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