How do I Find the Vector Coordinates to Solve for Trihedral Angle?

In summary, The problem is to find the coordinates of a unit vector C in 3D that has angles α and β with unit vectors A and B respectively, while the sum of α and β is equal to the given angle φ and the total sum of α, β, and φ is less than or equal to 360°. The solution involves using the dot product equations A.C = cos(α) and B.C = cos(β) and a third equation, the triple product A.BxC = V, where V2 = 1 + 2*cos(α)*cos(β)*cos(φ) – cos2(α) – cos2(β) – cos2(φ). This is known as the
  • #1
Julian1
2
0
Hi everyone,

Here's the problem I have.

Given two unit vectors A, B and angle φ between them. Find the coordinates (in 3D) of a unit vector C so that the angles between C and A,B be α and β respectively.
α + β => φ and α + β + φ <= 360°

It looks trivial to me and yet here I am asking for help:)

I have two equations from the dot products:
A.C = cos(α)
B.C = cos(β)
and I need a third one to solve the problem?

Thanks!Julian.
 
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  • #2
Re: find the vector coordinates

I have found the solution.

The third equation is the triple product.

A.BxC = V

where

V2 = 1 + 2*cos(α)*cos(β)*cos(φ) – cos2(α) – cos2(β) – cos2(φ)

Trihedral Angle | OPEN MIND
 

FAQ: How do I Find the Vector Coordinates to Solve for Trihedral Angle?

What is a vector?

A vector is a mathematical quantity that has both magnitude (size) and direction. It is often represented by an arrow pointing in the direction of the vector with its length representing the magnitude.

How do you find the coordinates of a vector?

To find the coordinates of a vector, you need to know the starting and ending points of the vector. Subtract the x coordinates of the starting point from the x coordinates of the ending point to find the x coordinate of the vector. Similarly, subtract the y coordinates of the starting point from the y coordinates of the ending point to find the y coordinate of the vector.

What is the difference between a position vector and a free vector?

A position vector is a vector that starts at the origin (0,0) and ends at a specific point. It is typically used to describe the position of an object in space. On the other hand, a free vector has no fixed starting point and can be translated anywhere in space without changing its magnitude or direction.

What are the rules for vector addition and subtraction?

When adding or subtracting vectors, the vectors must have the same number of dimensions. To add vectors, simply add the corresponding components of the vectors. To subtract vectors, subtract the corresponding components. The result will be a new vector with the same number of dimensions as the original vectors.

How do you find the magnitude and direction of a vector?

The magnitude of a vector can be found using the Pythagorean theorem. The square of the magnitude is equal to the sum of the squares of the individual components. To find the direction of a vector, you can use trigonometric functions such as sine and cosine to find the angle between the vector and a reference axis.

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