Consider the vectors a = 2 i + 3 j + 4 k and b = 3 i + 4 j + 5 k . Determine the angle between these vectors.

Step-by-step solution

The given vectors are a = 2 i + 3 j + 4 k and b = 3 i + 4 j + 5 k . The dot product of the vectors is: a b = (2)(3) + (3)(4) + (4)(5) = 6 + 12 + 20 = 38 The magnitudes of the vectors are: | a | = 2^2 + 3^2 + 4^2 = 4 + 9 + 16 = 29 | b | = 3^2 + 4^2 + 5^2 = 9 + 16 + 25 = 50 The angle between the vectors is given by: = a b | a | | b | Substituting the values: = 38 29 50 = 38 1450 = ⁻¹ ( 38 1450 )