Finding the intersection point of two 3D vectors?

In summary, the conversation discusses a problem involving a helicopter and an airplane travelling in different directions at different speeds. The goal is to determine the direction, coordinates, and time of interception between the two vehicles. The conversation mentions using the dot or cross product of the helicopter's vectors, but ultimately suggests using simultaneous equations to solve the problem.
  • #1
RandomStudent
2
0

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!
 
Physics news on Phys.org
  • #2
RandomStudent said:
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! :smile:

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! :wink:
 

1. What is a 3D vector?

A 3D vector is a mathematical representation of a direction and magnitude in three-dimensional space. It consists of three components (x, y, and z) and is typically denoted as <x, y, z>.

2. How do you find the intersection point of two 3D vectors?

To find the intersection point of two 3D vectors, you can use the cross product or dot product. The cross product results in a vector perpendicular to both vectors, which can be used to determine the intersection point. The dot product results in a scalar value that can also be used to determine the intersection point.

3. Can two 3D vectors intersect at more than one point?

No, two 3D vectors can only intersect at one point. This is because a 3D vector represents a direction and magnitude, and only one point can have the same direction and magnitude as two vectors.

4. What is the significance of finding the intersection point of two 3D vectors?

The intersection point of two 3D vectors is significant in many applications, such as computer graphics, physics, and engineering. It can be used to determine the position of an object in space, calculate the angle between two vectors, or solve systems of equations.

5. Are there any real-world examples of 3D vector intersections?

Yes, there are many real-world examples of 3D vector intersections. For instance, in GPS navigation systems, multiple satellites use 3D vectors to determine the intersection point of a user's location. In architecture and construction, 3D vectors are used to determine the intersection points of beams and columns in a building. In video games, 3D vectors are used to calculate the intersection of a player's line of sight with objects in the game world.

Similar threads

Unit Vector Problem: Find Point of Intersection
    • Calculus and Beyond Homework Help
Replies
2
Views
3K
Finding the point on the line of intersection between planes
    • Calculus and Beyond Homework Help
Replies
5
Views
2K
The Line of Intersection of Two Planes
    • Calculus and Beyond Homework Help
Replies
2
Views
1K
Finding the Vector Equation for the Intersection of Two Planes
    • Calculus and Beyond Homework Help
Replies
2
Views
1K
Find a vector tangent to the curve of intersection of two cylinders
    • Calculus and Beyond Homework Help
Replies
5
Views
8K
Finding intersection of vector subspaces
    • Calculus and Beyond Homework Help
Replies
3
Views
5K
Finding the point of intersection and distance of a vector
    • Calculus and Beyond Homework Help
Replies
1
Views
2K
Finding line of intersection of two planes
    • Calculus and Beyond Homework Help
Replies
25
Views
4K
Finding a tangent vector to the intersection of two surfaces
    • Calculus and Beyond Homework Help
Replies
3
Views
4K
Finding three planes which intersect a point with lines
    • Calculus and Beyond Homework Help
Replies
1
Views
2K

Hot Threads

Recent Insights

Back
Top