Final answer:
To find the angle between two vectors, you can use the dot product formula: a · b = |a| |b| cos(θ). Given that a = (1, -4, 1) and b = (0, 5, -5), the angle between the vectors is approximately 113.5 degrees.
Step-by-step explanation:
To find the angle between two vectors, you can use the dot product formula:
a · b = |a| |b| cos(θ)
Given that a = (1, -4, 1) and b = (0, 5, -5), the dot product of a and b is:
a · b = (1)(0) + (-4)(5) + (1)(-5) = -20
The magnitudes of the vectors are |a| = sqrt(12 + (-4)2 + 12) = sqrt(18) and |b| = sqrt(0^2 + 5^2 + (-5)^2) = sqrt(50).
Substituting these values into the formula, we have:
-20 = sqrt(18) sqrt(50) cos(θ)
Solving for θ, we get:
θ = acos(-20 / (sqrt(18) sqrt(50)))
Calculating this value gives us:
θ ≈ 113.5 degrees