Today we'll start with cross product problems from the definintion. Then let's get to plane problems.
- Use the definition of the cross product to explain why $2\hat i \times 3\hat j=6\hat k$.
- Does $\hat j \times \hat k$ equal $\hat i$ or $-\hat i$. Explain. What about $\hat i\times \hat k$? Use the right hand rule to make your decision.
- A parallelogram has two edges given by $\vec u = (3,1,-2)$ and $\vec v=(4,0,3)$. Find the area of the parallelogram.
- Consider the three points $P=(2,0,0)$, $Q=(0,3,0)$, and $R=(0,0,5)$.
- Give a vector $\vec n$ that is orthogonal to both $\vec {PQ}$ and $\vec{PR}$.
- Give the area of the triangle whose vertices are $P$, $Q$, and $R$.
- The three points above lie on a plane. Let $S=(x,y,z)$ be any point on the plane. State the vector $\vec {PS}$ and the angle between $\vec n$ and $\vec {PS}$.
- Give an equation of the plane that passes through $P$, $Q$, and $R$.
- Rewrite your equation in the form $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$.
- Repeat the previous problem with 3 other points.
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