Final answer:
To find the angle theta between the vectors u and v, we can use the dot product formula and the magnitudes of the vectors. First, calculate the dot product of the two vectors by multiplying their corresponding components and adding them together. Next, find the magnitudes of the vectors. Finally, use the dot product and magnitudes to calculate the angle theta in radians and degrees.
Step-by-step explanation:
To find the angle theta between the vectors in radians and in degrees, we can use the dot product formula and the magnitude of the vectors. First, calculate the dot product of the two vectors by multiplying their corresponding components and adding them together: u·v = (1)(2) + (0)(-6) + (-3)(1) = 2 - 0 - 3 = -1. Next, find the magnitudes of the vectors: |u| = √(1² + 0² + (-3)²) = √10 and |v| = √(2² + (-6)² + 1²) = √41. Finally, use the formula cos(theta) = u·v / (|u| |v|) to find the cosine of the angle theta: cos(theta) = -1 / (√10 √41). To find theta in radians, use the inverse cosine function: theta = cos^(-1)(-1 / (√10 √41)). To find theta in degrees, convert the radians to degrees by multiplying by (180/π).