Question

A robot is put into a maze, it can only go N, E, S, and West. The value i represents the north, and the magnitude is equal to 1. I have figured out that N= i, East= 1, South= -i, and West= -1. When programming the robot, let the complex number d represent the direction the robot is facing. As the robot changes direction, the value of d will also change; so, the value of d is dependent on where the robot is in the maze. When the robot makes a turn, it would be useful to have an operation to perform on d to represent this turn. This is because after making a turn, the new value of d will depend on the old value of d.

Complete the table for the new values of d if the robot is turning left or right. Then determine an expression in terms of d that will give the new position if the robot turns left and another expression if the robot turns right.

Answer 1

Displacement problems in physics are solved using vector addition and trigonometry to divide into east and north components. The total displacement and direction are calculated from the magnitude and direction of the resultant vector.

Step-by-step explanation:

To solve problems involving displacement using analytical techniques in physics, we typically break down the displacement into its components, such as east and north components. Let's consider a few examples to explain how to calculate the total displacement from the starting point.

For the scenario of walking 12.0 m in a direction 20° west of north, followed by 20.0 m in a direction 40° south of west, we can represent these displacements using vectors. We would find the components of each leg of the trip along the north-south and east-west axes first. By using trigonometry, we can calculate these components and then add the vectors to find the total displacement from the start point. The magnitude of this resultant vector gives us how far the person is from the starting point, and by calculating the direction of this vector, we can determine the compass direction to the final position.

In the specific example of walking 18.0 m straight west and then 25.0 m straight north, the displacement vectors are perpendicular to each other. We can use the Pythagorean theorem to find the total distance from the starting point, which is the magnitude of the resultant vector. To find the compass direction of this resultant displacement, we can use the arctangent function to find the angle of the displacement relative to the north or east axis.