Calculus III vector problem Can someone help explain the solution in detail?
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 (4,0) . If he began walking in the direction of the vector V= 5i+1j and changes direction only once, when he turns at a right angle, what are the coordinates of the point where he makes the turn.
2 Responses
w
13 Mar 2010
nle
13 Mar 2010
let point A( 2, -3 ) , point B ( 4,0)
M ( x, y ) is where he makes the turn.
He starts at A , then vector AM = k [ 5,1 ] where k is a real number
remember [5,1] is the components of vector V.
Now component of vector BM = [ x - 4, y ]
Since the dot product of AM_vec and BM_vec =0 then
5k (x -4) + k y = 0 or y = -5x + 20 ( eq1)
Now write the equation of the circle with diameter A and B
It is ( x – 3) ^2 + ( y + 3/2) ^ 2 = 13/4 ( eq2 )
Solve simultaneously eq 1 and eq2 then x = 9/2 ( discard x =4 )
then M [ 9/2 , -5/2 ]
Im not sure , but I think
(y–3)/(x-2) = (y+3)/(x+2)= -(x-4)/(y-0),
since its perpendicular and
(y–3)/(x-2)=(y+3)/(x-2) = 1/5
because of vector,
so solving
(x,y) = (9/2, -5/2)