Chapter 2, Problem 36RP

a

To determine

To Find : The coordinates of B for given condition.

Answer to Problem 36RP

  $$ The coordinates of B $=\text{ (- 1 , - 2 )}$

Explanation of Solution

Given: We are given that with respect to Origin, point $A\text{=(1,2)}$ is rotated $100°$ clockwise, then $80°$ counterclockwise, then $210°$ clockwise ,and finally $50°$ counterclockwise to point B.

  

Concept Used:

Using rule of clockwise and counterclockwise rotation where clockwise stands for negative and counterclockwise stands for positive movement of angles.

Calculation:

As shown in the figure the point O is showing origin and A is representing the location of coordinates $(1,\text{ 2 )}$.Also we have been given the movement of point A clockwise and counterclockwise

Applying the rule of clockwise and counterclockwise rotation on given information we can break down the given information and write following points:

  $\begin{array}{l}1)\text{ 100}°\text{ clockwise = - 100}°\\ \\ \text{2) 80}°\text{ counterclockwise = + 80}°\\ \\ \text{3) 210}°\text{ clockwise = - 210}°\\ \\ \text{4) 50}°\text{ counterclockwise = +50}°\end{array}$

As we are given after all these movements point B is obtained. Thus, to know the position of point B adding all the movements:

  $\begin{array}{l}=\text{ - 100 }°\text{ + 80}°\text{ - 210}°\text{ + 50}°\\ \\ \text{= -180}°\end{array}$

Thus, point B is at the clockwise rotation of $180°$ from A. Which is shown by following graph.

  

So the coordinates of point B will be $(\text{ - 1, -2 )}$ as it is at $180°$ clockwise rotation from A.

Conclusion:

The coordinates of B $=\text{ (- 1 , - 2 )}$

b

To determine

To Find: The rotation after which the point is in first quadrant again.

Answer to Problem 36RP

The point was in first quadrant after second rotation.

Explanation of Solution

Given: We are given that with respect to Origin, point $A\text{=(1,2)}$ is rotated $100°$ clockwise, then $80°$ counterclockwise, then $210°$ clockwise ,and finally $50°$ counterclockwise to point B.

  

Concept Used:

Using rule of clockwise and counterclockwise rotation where clockwise stands for negative and counterclockwise stands for positive movement of angles.

Calculation:

As shown in the figure the point O is showing origin and A is representing the location of coordinates $(1,\text{ 2 )}$.Also we have been given the movement of point A clockwise and counterclockwise

Now as the point A is in first quadrant to know that when again it will come in same quadrant applying the rule of clockwise and counterclockwise rotation on given information, we can break down the given information and write following points:

  $\begin{array}{l}1)\text{ 100}°\text{ clockwise = - 100}°\\ \end{array}$

After this movement the point will move in the fourth quadrant.

  $\text{2) 80}°\text{ counterclockwise = + 80}°$

After this movement the point will come back in the first quadrant.

After this for next two rotation it will shift to third quadrant and stay there.

So, after second rotation the point again will come in first quadrant

Conclusion:

The point was in first quadrant after second rotation