Gandalf the Grey started in the Forest of Mirkwood at a point with coordinates (−2,3) and arrived in the Iron Hills at the point with coordinates (−1,7). If he began walking in the direction of the vector v=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.
I'm stuck after doing the dot product. Thank you
no need for cross or dots I think
turn at (a,b)
what are the angles steered counterclockwise from x axis (I do ships but will try to stick to the math angles :)
?
first leg
tan Theta = 2/3 so theta = 37.4 degrees
now what is the angle from start to finish
delta y = 7 - 3 = 4
delta x = -1 +2 = 2
so tan angle = 4/2 = 2 so angle = 70.5 degrees
we are there.
in our triangle we have 70.5- 37.4 = 33 deg at start and 90 deg at the left turn
so 90 - 33 = 57 degrees up at the finish
we know the hypotenuse sqrt[(7-4)^2 +1^2] = sqrt 10
so
cos 33 = length = that first leg / sqrt 10
you now have the length of the first leg and the direction is 37.4 degrees above East so you can find the x y changes easy from (-2,3)
2. You start a search for a buried object by marking the center of a field as (0, 0), with coordinates giving distance in yards. Coordinates to the north of east are positive, and coordinates to the south or west are negative. You find nothing at (-10, 6), so you try a likely looking spot 3 yards to the east and 12 yards to the south of the first spot. What are the coordinates of the second spot?
The first spot has coordinates (-10, 6). Moving 3 yards east from this point gives us (-10 + 3, 6) = (-7, 6). Moving 12 yards south from here gives us (-7, 6 - 12) = (-7, -6). Therefore, the second spot has coordinates (-7, -6).
To find the coordinates where Gandalf makes the turn, we need to determine the point at which his direction vector is perpendicular to the vector between his starting point and the turning point.
Let's break down the problem step by step:
1. Find the vector between the starting point and the turning point:
Vector u = (turning point) - (starting point)
u = (-1, 7) - (-2, 3)
= (-1, 4)
2. The dot product of two vectors is given by the formula:
Dot Product = u · v = |u||v|cos(θ)
Since Gandalf turns at a right angle, cos(θ) = 0.
3. Solve the dot product equation to find the scalar value t:
u · v = |u||v|cos(θ)
(-1)(3) + (4)(2) = sqrt((-1)^2 + 4^2) * sqrt(3^2 + 2^2) * 0
-3 + 8 = sqrt(17) * sqrt(13) * 0
5 = 0
Since 5 = 0 is not true, it means there is no value of t that satisfies the equation. This indicates that Gandalf does not make a turn at a right angle. However, if you made a mistake and Gandalf does turn at a right angle, please double-check your calculations.
If you have any further questions or need additional assistance, please let me know!