Final answer:
To find the angle between two vectors A= (2, 3, 6) and B = (1, -1, 1), calculate their dot product, then use this to find the cosine of the angle, and lastly take the inverse cosine to get the angle.
Step-by-step explanation:
To find the angle between two vectors, we use the dot product formula and the magnitudes of both vectors. The dot product of vectors A and B is A · B = Ax × Bx + Ay × By + Az × Bz. To find the angle θ between vectors A = (2, 3, 6) and B = (1, -1, 1), we first calculate the dot product A · B = (2 × 1) + (3 × -1) + (6 × 1). Then, find the magnitudes |A| and |B| and use the formula cosθ = (A · B) / (|A| × |B|) to calculate the angle.
First, A · B = 2 × 1 + 3 × (-1) + 6 × 1 = 2 - 3 + 6 = 5.
The magnitude of vector A is |A| = √(2^2 + 3^2 + 6^2) = √(4 + 9 + 36) = √49 = 7.
The magnitude of vector B is |B| = √(1^2 + (-1)^2 + 1^2) = √(1 + 1 + 1) = √3.
Therefore, cosθ = 5 / (7 * √3). To find the angle, take the inverse cosine of this value, θ = cos^-1(5 / (7 * √3)).