Ū= (5,8) Find the direction angle of ū.
Enter your answer as an angle in degrees between 0 and 360° rounded to the nearest hundredth.

Assuming the angle made by a vector refers to the angle it makes with the positive horizontal axis, take the dot product of u with the vector (1, 0), and recall that

a • b = ||a|| ||b|| cos(θ)

where θ is the angle made by the vectors a and b. Then

u • (1, 0) = ||u|| ||(1, 0)|| cos(θ)

We have

u • (1, 0) = 5×1 + 8×0 = 5

||u|| = √(5² + 8²) = √89

||(1, 0)|| = √(1² + 0²) = √1 = 1

so that

5 = √89 cos(θ)

cos(θ) = 5 / √89

θ = arccos(5 / √89) ≈ 57.99°

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