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vec(u)=(-2,5) Find the direction angle of vec(u). Enter your answer as an angle in degrees between 0^(@) and 360^(@) rounded to the nearest hundredth. theta=◻" 。 "

u⃗=(−2,5) \vec{u}=(-2,5) \newlineFind the direction angle of u⃗ \vec{u} . Enter your answer as an angle in degrees between 0∘ 0^{\circ} and 360∘ 360^{\circ} rounded to the nearest hundredth.\newlineθ=□∘ \theta=\square^{\circ}

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Q. u⃗=(−2,5) \vec{u}=(-2,5) \newlineFind the direction angle of u⃗ \vec{u} . Enter your answer as an angle in degrees between 0∘ 0^{\circ} and 360∘ 360^{\circ} rounded to the nearest hundredth.\newlineθ=□∘ \theta=\square^{\circ}
  1. Use Arctangent Function: To find the direction angle of the vector u⃗=(−2,5) \vec{u} = (-2,5) , we need to use the arctangent function, which gives us the angle whose tangent is the ratio of the y-coordinate to the x-coordinate of the vector. The formula for the direction angle θ \theta is:\newlineθ=arctan⁡(yx) \theta = \arctan\left(\frac{y}{x}\right) \newlinewhere x x and y y are the x-coordinate and y-coordinate of the vector, respectively.
  2. Plug Coordinates into Formula: First, we plug the coordinates of u⃗ \vec{u} into the formula:\newlineθ=arctan⁡(5−2) \theta = \arctan\left(\frac{5}{-2}\right)
  3. Calculate Arctangent: Calculating the arctangent of 5−2 \frac{5}{-2} gives us an angle in radians. We need to convert this angle to degrees and make sure it is in the range of 00° to 360360°. Since the vector is in the second quadrant (negative x-coordinate and positive y-coordinate), the direction angle θ \theta will be 180° 180° minus the angle we find.
  4. Convert to Degrees: Using a calculator, we find that:\newlineθ=arctan⁡(5−2)≈arctan⁡(−2.5) \theta = \arctan\left(\frac{5}{-2}\right) \approx \arctan(-2.5) \newlineθ≈−68.1985905° \theta \approx -68.1985905° \newlineSince the angle is negative, we add 180180° to find the direction angle in the second quadrant:\newlineθ=180°−(−68.1985905°) \theta = 180° - (-68.1985905°) \newlineθ=180°+68.1985905° \theta = 180° + 68.1985905° \newlineθ≈248.1985905° \theta \approx 248.1985905°
  5. Round to Nearest Hundredth: Finally, we round the direction angle to the nearest hundredth:\newlineθ≈248.20° \theta \approx 248.20°

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