Question

A news helicopter is descending along the helix (sin(πt), cos(πt), 10 − t). At time t = 5 the crew turns on a powerful head light shining straight ahead in the direction of the velocity vector. What spot on the ground, i.e. what point on the xy-plane, does this beam of light hit?

Answer 1

1. Begin by finding the velocity vector v(t) - to do so, differentiate your position function r(t)=(sin(pi*t),cos(pi*t),10-t).
2. Form parametric equations of the tangent line to the helix at t=5 - this is the line that the beam of light will follow. This line passes through the point r(5) and has a direction vector v(5).
3. Set the z-coordinate of the line obtained in part 2 equal to zero, and solve for t. Then substitute the resulting t value into the line equation x and y-coordinates.

Answer 2

r(t) = ( sin(πt), cos(πt), 10-t ) >> r(5) = (0, -1, 5)
v(t) = ( -π(cos(πt)), -π(sin(πt)), -1) >> v(5) = (-5π, 0, -1)
sorry, i'm pretty horrible at math, but am i supposed to set 10-t = 0 and solve for t which =10? the answer to the question is supposed to be (-5π, -1). thank you so much!

Answer 3

v(t) should be ( π(cos(πt)), -π(sin(πt)), -1) so v(5) = (-π, 0, -1)

Parametric equations of the tangent line passing through the point (a,b,c) in the direction of a vector <d,e,f> are
x = a + d*t, y = b+ e*t, z = c + f*t
Use the numbers from your r(5) and v(5) vectors as a,b,c,d,e,f, then proceed to step 3. from my previous message...

Answer 4

Ohhhh I see. Gosh I made so many errors. Thanks a bunch :D