Final answer:
The angle between the vectors A = 2î + 3ᴇ - ᴇᴥ and B = 4î + 6ᴇ - 2ᴇᴥ is found using the dot product and magnitudes of the vectors. The cosine of the angle is 1, indicating the vectors are parallel, and thus the angle between them is 0 degrees.
Step-by-step explanation:
The question asks to find the angle between two vectors, A and B. The dot product and magnitudes of the vectors are used to calculate the cosine of the angle between them. Given vectors A = 2î + 3ᴇ - ᴇᴥ and B = 4î + 6ᴇ - 2ᴇᴥ, the dot product A ⋅ B is:
(2)(4) + (3)(6) + (-1)(-2) = 8 + 18 + 2 = 28
The magnitudes of vectors A and B are |A| = √(2² + 3² + (-1)²) = √(4 + 9 + 1) = √14 and |B| = √(4² + 6² + (-2)²) = √(16 + 36 + 4) = √56 respectively.
The cosine of the angle theta between vectors A and B is then given by cos(theta) = (A ⋅ B) / (|A| ⋅ |B|). Substituting the values, we get:
cos(theta) = 28 / (√14 ⋅ √56)
cos(theta) = 28 / (√784)
cos(theta) = 28 / 28
cos(theta) = 1
Since the cosine of the angle is 1, the angle between vectors A and B is 0 degrees. Therefore, the answer is (a) 0°.