Chapter 9: Polar Coordinates, Vectors

In Problems 8 and 9, test the polar equation for symmetry with respect to the pole, the polar axis, and the line $\theta =\frac{\pi }{2}$.

$r=5\sin \theta {\cos }^2\theta$

Describe the set of points $\left(x,y,z\right)$defined by the equation(s).

$z=2$

Removing a Stump A farmer wishes to remove a stump from a field by pulling it out with his tractor. Having removed many stumps before, he estimates that he will need $6$tons ($12,000$pounds) of force to remove the stump. However, his tractor is only capable of pulling with a force of $7000$pounds, so he asks his neighbor to help. His neighbor’s tractor can pull with a force of $5500$pounds. They attach the two tractors to the stump with a ${40}^{\circ }$-angle between the forces as shown in the figure.

1. Assuming the farmer’s estimate of a needed $6$-ton force

   is correct, will the farmer be successful in removing the

   stump? Explain.

2. Had the farmer arranged the tractors with a ${25}^{\circ }$angle between the forces, would he have been successful in removing the stump?Explain.

Explain why the vertical-line test used to identify functions

in rectangular coordinates does not work for equations

expressed in polar coordinates.

Static EquilibriumShow on the following graph the force needed for the object at P to be in static equilibrium.

Open the Vectors applet. Draw the directed line segment from $P_1=(1,2)$to $P_2=(5,4)$. Then draw the position vector $\mathbf{v}=\overrightarrow{P_1{P}_2}$.

Open the Vectors applet. Suppose $\mathbf{v}=2\mathbf{i}+3\mathbf{j}$and $\mathbf{w}=4\mathbf{i}-3\mathbf{j}$.

1. Draw v and then use the red vector to show $3\mathbf{v}$.
2. Use the red vector to show -v.
3. Draw v and w. Then use the red vector to show $\mathbf{v}+\mathbf{w}$.
4. Use the red vector to show $\mathbf{v}-\mathbf{w}$.
5. Use the red vector to show $2\mathbf{v}+\mathbf{w}$.

Explain in your own words what a vector is. Give an example of a vector.

Write a brief paragraph comparing the algebra of complex numbers and the algebra of vectors.

Explain the difference between an algebraic vector and a position vector.