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Calculus 3: test 1

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1. Vectors $a$ and $b$ are given below. Copy the picture and illustrate graphically (a) $a+b,$ (b) $a-b,$ (c) $|a|,$ (d) $a\times b$.

Two vector a and b.png

2. Find the angle between the vectors $<1,1,1>$ and $\mathbf{i}$. Don't simplify.

3. Find the plane through the origin perpendicular to the line from $(1,0,0)$ and $(0,1,1)$.

4. Plot and describe the curve \[x(t)=|\sin t|,y(t)=|\cos t|,z=t.\]

5. Find the line tangent to the curve \[ \mathbf{r}(t)=<t^{5},t^{4},t^{3}> \] at point $(1,1,1)$.

6. Find the unit normal vector of the curve \[ \mathbf{r}(t)=3t\mathbf{i}+2t^{2}\mathbf{j}. \]

7. Suppose a ball in thrown horizontally at speed $w$ feet per second by a person $h$ feet tall. At what speed will the ball hit the ground?