Final answer:
To find the angles of a triangle with given vertices, calculate the vectors representing the sides of the triangle, find the dot products, use the cosine rule, calculate the magnitude of each vector, and then determine the angles using the inverse cosine function.
Step-by-step explanation:
To find the three angles of a triangle with given vertices (1,1,1), (1,−5,2), and (−6,2,7), we can use the concept of vectors and the dot product to determine the angle between them. A triangle with vertices in a three-dimensional space is still subject to the rule that the sum of its angles equals 180 degrees. We'll first find the vectors that represent the sides of the triangle, then calculate the dot products and finally use the cosine rule to find the angles.
First, calculate the vectors for the sides of the triangle:
- Vector AB = B - A
- Vector BC = C - B
- Vector CA = A - C
Next, calculate the dot product of these vectors to find the cosine of the angle between them:
- cos(θ) = (Vector AB ⋅ Vector AC) / (|AB| |AC|)
- cos(β) = (Vector BC ⋅ Vector AB) / (|BC| |AB|)
- cos(γ) = (Vector CA ⋅ Vector BC) / (|CA| |BC|)
Then, calculate the magnitude (length) of each vector using the formula:
|Vector| = √(x² + y² + z²)
Now, the angle in degrees can be found using the inverse cosine:
- θ = cos⁻¹(cos(θ))
- β = cos⁻¹(cos(β))
- γ = cos⁻¹(cos(γ))
Now, we have all three angles of the triangle.