The expression that will tell you the direction the robot is going will be ((i)²¹*(-i)²²)d i.e. right to the initial direction. And the order does not matter for the direction it is finally facing.
What is rotation in complex plane?
When in the complex plane, the line is rotated clockwise, then the line is multiplied by -i and when the line is rotated anti-clockwise, then the line is multiplied by i.
A right turn is the clockwise rotation of the direction vector by the angle of 90°. It is equivalent to multiplying the complex number by -i.
A left turn represents a counterclockwise rotation of the direction vector by the angle of 90°. It is equivalent to multiplying the complex number by i.
Here assume the initial direction vector of the robot is d
the robot is turned to the left 21 times. so now the direction vector will be di²¹. Then it is rotated right 22 times, now the direction vector will be (di²¹)*(-i)²²=d((i)²¹*(-i)²²).
So, now the direction that the robot is going= d((i)²¹*(-i)²²)= di²¹⁺²²=di⁴³= d(-i)= -di
means the robot is the right of the initial direction.
Now the expression that shows the direction that the robot is going is d((i)²¹*(-i)²²).
The order does not matter as the direction will always be the same irrespective of the order.
Therefore The expression that will tell you the direction the robot is going will be ((i)²¹*(-i)²²)d i.e. right to the initial direction. And the order does not matter for the direction it is finally facing.
Learn more about the complex plane
here: https://brainly.com/question/16835201
#SPJ2
