Tricky minimum distance vector problem

• Lucy Yeats
In summary, Lucy Yeats was trying to solve a problem involving finding the value of μ at the point of intersection and calculating the distance. She found that she needed to find μ first, and then used the distance formula to find the distance.

Homework Statement

The line intersects a plane at an angle alpha=2pi/3. The line is defined by rn, and the plane by r.m=0, with n and m unit vectors. Calculate the shortest distance from the plane to the point on the line with μ=2.

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The Attempt at a Solution

I want to find the value of μ at the point of intersection. Then it will be easy to find the distance, as d=(2-μ at intersection)sin(2pi/3). However, I don't know how to find μ. Any hints?

I'm still stuck. Any ideas would be much appreciated!

Hi Lucy Yeats!

To find μ at the point of intersection you need to substitute the equation of the line into the equation of the plane.
That is (μn).m=0. From this it follows that μ=0.

That's funny!
Yes, the origin is on the line and the origin is also in the plane.

Btw, the regular way to calculate the distance of a point to a plane, is to project any vector from the plane to the point, onto the unit normal vector.
You can do this with the dot product.

If n and m are perpendicular, surely mu could be anything? How do you know that they aren't perpendicular?

I'm guessing that you have to use the angle 2pi/3 in some way?

Thanks for helping, btw!

Lucy Yeats said:
If n and m are perpendicular, surely mu could be anything? How do you know that they aren't perpendicular?

Yes.
What would happen (geometrically speaking) with the point of intersection if n and m are perpendicular?
Lucy Yeats said:
I'm guessing that you have to use the angle 2pi/3 in some way?

Thanks for helping, btw!

Yes.
But first you need the formula for the distance of a point to a plane:
distance = (any vector from plane to point) . (normal vector of plane)

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Actually, your distance formula d=(2-μ at intersection)sin(2pi/3) also works.

Just fill in the μ we just found for the intersection.
Btw, did you know that n.m=sin(2pi/3)?

Ah, I see, they can't be perpendicular because then the angle wouldn't be 2pi/3. so mu has to be zero.

Brilliant, thank you very much!

You're welcome!

1. What is the "tricky minimum distance vector problem"?

The tricky minimum distance vector problem (TMDVP) is a mathematical optimization problem that involves finding the shortest path between two points in a network. It is commonly used in computer science and telecommunications to determine the most efficient routing of data packets.

2. How does the TMDVP differ from other minimum distance vector problems?

The TMDVP differs from other minimum distance vector problems in that it takes into account additional constraints, such as limited resources or variable costs, which can make finding the optimal solution more challenging.

3. What are some real-world applications of the TMDVP?

The TMDVP has many real-world applications, including in transportation and logistics, where it is used to optimize delivery routes. It is also used in telecommunications to improve network performance.

4. What are some common algorithms used to solve the TMDVP?

Some common algorithms used to solve the TMDVP include Dijkstra's algorithm, Bellman-Ford algorithm, and the Floyd-Warshall algorithm. These algorithms use different approaches to find the shortest path and can be tailored to fit specific constraints and scenarios.

5. What are the challenges in solving the TMDVP?

The TMDVP can be challenging to solve because it involves finding the optimal solution among a large number of possible routes, which can be computationally intensive. In addition, the constraints and variables involved in the problem can make it difficult to find an efficient and accurate solution.