The triangle is a simplest polygon for which finding of sizes of corners in the known parameters (to lengths of the parties, radiuses of the entered and circumscribed circles, etc.) there are several formulas. However the tasks demanding calculation of corners in tops of a triangle which is placed in some spatial system of coordinates often meet.
Instruction
1. If the triangle is set by coordinates of all three tops (X ₁, Y ₁, Z ₁, X ₂, Y ₂, Z ₂ and X ₃, Y ₃, Z ₃), then begin with calculation of lengths of the parties forming that corner of a triangle (α) which size interests you. If any of them to complete its length to a rectangular triangle in which the party will be a hypotenuse, and its projections to two axes of coordinates - legs then it is possible to find on Pythagorean theorem. Lengths of projections will be equal to the difference of coordinates of the beginning and the end of the party (i.e. two tops of a triangle) on the corresponding axis, so, length can be expressed as a square root from the sum of squares of differences of such coordinate couples. For three-dimensional space the corresponding formulas of two parties of a triangle can be written down so: √ ((X -X ₂)² + (Y -Y ₂)² + (Z -Z ₂)²) and √ ((X -X ₃)² + (Y -Y ₃)² + (Z -Z ₃)²).
2. Use two formulas of a scalar product of vectors - in this case vectors with the general beginning are the parties of a triangle forming the calculated corner. One of formulas expresses a scalar product through their lengths received by you on the previous step and a cosine of the angle between them: √ ((X -X ₂)² + (Y -Y ₂)² + (Z -Z ₂)²) * √ ((X -X ₃)² + (Y -Y ₃)² + (Z -Z ₃)²) * cos(α). Another - through the sum of works of coordinates on the corresponding axes: X *X ₃ + Y *Y ₃ + Z *Z ₃.
3. Equate these two formulas and express a cosine of a required corner from equality: cos(α) = (X *X ₃ + Y *Y ₃ + Z *Z ₃) / (√ ((X -X ₂)² + (Y -Y ₂)² + (Z -Z ₂)²) * √ ((X -X ₃)² + (Y -Y ₃)² + (Z -Z ₃)²)). The trigonometrical function determining corner size in degrees by value of its cosine is called an arccosine - use it for record of a final version of a formula of finding of a corner on three-dimensional coordinates of a triangle: α = arccos ((X *X ₃ + Y *Y ₃ + Z *Z ₃) / (√ ((X -X ₂)² + (Y -Y ₂)² + (Z -Z ₂)²) * √ ((X -X ₃)² + (Y -Y ₃)² + (Z -Z ₃)²))).