Question

Gandalf the grey started in the forest of mirkwood at a point pp with coordinates (−2,1)(−2,1) and arrived in the iron hills at the point qq with coordinates (-1, 6). if he began walking in the direction of the vector v=3i+2jv=3i+2j and changes direction only once, when he turns at a right angle, what are the coordinates of the point where he makes the turn?

Answer 1

Gandalf makes a right-angle turn while traversing from point (-2,1) to (-1,6). Without distances, we cannot find the exact coordinates of the turn but can infer the direction after the turn should be perpendicular to his initial direction vector 3i+2j. We deduce the direction is -2i+3j, pointing towards the endpoint, but the coordinates remain indeterminate.

Step-by-step explanation:

The student wishes to determine the coordinates of the point where Gandalf makes a turn, given that he starts walking in the direction of the vector v=3i+2j and changes direction only once by turning at a right angle. Gandalf starts at point p with coordinates (-2,1) and ends at point q with coordinates (-1,6). The solution involves breaking down the path from p to q into two vectors that are perpendicular to each other and finding their point of intersection.

First, we consider Gandalf's initial direction vector v=3i+2j. Since he eventually turns 90 degrees, the new direction vector after the turn would be perpendicular to v. To find a vector that is perpendicular to v, we can take its components and switch their places, changing the sign of one of them, resulting in 2i-3j or -2i+3j. To reach the end point q, Gandalf must have walked along a vector parallel to one of these two directions.

The question now becomes which of the two perpendicular direction vectors leads to point q. By checking, we can see that starting from (-2,1) and moving in the direction of 2i-3j will not lead towards the endpoint q, but moving in the direction of -2i+3j will. Hence, Gandalf must have turned and walked in the direction of the vector -2i+3j. By following this vector, we can find the intersection point that lies on the straight path between p and q.

Unfortunately, without additional information about the distances traveled in each direction, the exact coordinates where Gandalf makes the turn cannot be determined from the information provided. We need more data, such as the total distance traveled or the distance traveled in a specific direction before the turn, in order to calculate the exact turning point's coordinates.

Answer 2

travel direction 3i + 2j, slope = 2/3.
We have slope and a point (-2,1) this gives us a line equation:
LINE BEFORE THE TURN, EQUATION: y = (2/3)x + 1

The point of intersection of the two lines:
1) through (-2,1) with slope 2/3 and;
2) through (-1,6) with slope 3/2.