Final answer:
To find the angle between the vectors u = (4, 2, 0) and v = (−2, 4, 4), we can use the dot product formula. The dot product of u and v is 0, and using this value, we can calculate the cosine of the angle between the vectors. The cosine of the angle is 0, which means the angle is 90 degrees or pi/2 radians.
Step-by-step explanation:
To find the angle between two vectors, we can use the dot product formula: \( \mathbf{u} \cdot \mathbf{v} = \| \mathbf{u} \| \| \mathbf{v} \| \cos \theta \).
Given that \( \mathbf{u} = (4, 2, 0) \) and \( \mathbf{v} = (-2, 4, 4) \), the dot product is \( \mathbf{u} \cdot \mathbf{v} = (4)(-2) + (2)(4) + (0)(4) = -8 + 8 + 0 = 0 \).
Therefore, \( \cos \theta = \frac{ \mathbf{u} \cdot \mathbf{v} }{ \| \mathbf{u} \| \| \mathbf{v} \| } = \frac{0}{ \sqrt{4^2 + 2^2 + 0^2} \sqrt{(-2)^2 + 4^2 + 4^2} } = 0 \).
Since \( \cos \theta = 0 \), the angle between the vectors is \( \theta = 90^\circ \) or \( \frac{\pi}{2} \) radians.