Finding the intersection point of two 3D vectors?

[RandomStudent]

Homework Statement

**NOTE: One coordinate unit = 1000 feet. Also, the helicopter and the airplane are TWO SEPARATE moving vehicles.**

At noon (12:00 PM), a helicopter is observed from point A (7, 0, 0) in the direction of vector -4i + 2j + 5k, and simultaneously from point B (0, 4, 0.25) in the direction of vector 3i - 2j + 4.75k. One minute later (12:01 PM), the helicopter is observed from A in the direction of vector 6i + 7j + 5.005k and from B in the direction 13i + 3j + 4.755k.

At 12:10 PM a different airplane is flying at the point Q (0, 0, 1) meaning that it is 1000 feet above the origin O. This airplane is traveling at 200 mph.

Assuming that the helicopter continues to travel in a fixed direction at a fixed rate, (1) IN WHICH DIRECTION (vector) should the airplane fly (from point Q) in order to intercept the helicopter? (2) WHERE (coordinate) and at (3) WHAT TIME will this interception occur?

Homework Equations

None except perhaps cross product? V1 = A1i + B1j + C1k. V2 = A2i + B2j + C2k. V1 x V2 = matrix of A1, B1, C1 and A2, B2, C2

The Attempt at a Solution

I'm not sure if I should find the dot or cross product of the helicopter's vectors. I was also wondering if parametrizing those vectors and points would help at all, because I did that but didn't know how to proceed...

Thanks in advance for any help or pointers!

 

[tiny-tim]

At noon (12:00 PM), a helicopter is observed from point A (7, 0, 0) in the direction of vector -4i + 2j + 5k, and simultaneously from point B (0, 4, 0.25) in the direction of vector 3i - 2j + 4.75k. One minute later (12:01 PM), the helicopter is observed from A in the direction of vector 6i + 7j + 5.005k and from B in the direction 13i + 3j + 4.755k.
…
None except perhaps cross product? V1 = A1i + B1j + C1k. V2 = A2i + B2j + C2k. V1 x V2 = matrix of A1, B1, C1 and A2, B2, C2
…
I'm not sure if I should find the dot or cross product of the helicopter's vectors. I was also wondering if parametrizing those vectors and points would help at all, because I did that but didn't know how to proceed...

Hi RandomStudent!

The obvious way is just to work out where the lines from A and B intersect (yes, I suppose you could call that using parameters) …

nothing fancy like dot or cross products, just good old-fashioned simultaneous equations!