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vec(u)=(9,-6) 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=(9,−6) \vec{u}=(9,-6) \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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Inverses of sin, cos, and tan: degreesright chevron icon

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Q. u⃗=(9,−6) \vec{u}=(9,-6) \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. Calculate Ratio: To find the direction angle of the vector u⃗=(9,−6) \vec{u} = (9, -6) , we need to calculate the angle that the vector makes with the positive x-axis. The direction angle θ \theta can be found using the arctangent function, which is the inverse of the tangent function. The tangent of the angle is the ratio of the y-coordinate to the x-coordinate of the vector. However, since the y-coordinate is negative and the x-coordinate is positive, the vector lies in the fourth quadrant, and we must add 360360 degrees to the arctangent of the ratio to get the angle in the range [00, 360360) degrees.
  2. Find Arctangent: First, calculate the ratio of the y-coordinate to the x-coordinate of the vector u⃗ \vec{u} :\newlinetan⁡(θ)=−69=−23 \tan(\theta) = \frac{-6}{9} = \frac{-2}{3} .
  3. Calculate Angle: Next, use the arctangent function to find the angle θ \theta that corresponds to this ratio. Remember that the arctangent function will give us an angle in the range (−90∘,90∘)(-90^{\circ}, 90^{\circ}), so we need to adjust it for the fourth quadrant.\newlineθ=arctan⁡(−23) \theta = \arctan\left(\frac{-2}{3}\right) .
  4. Adjust for Quadrant: Using a calculator, we find that:\newlineθ≈arctan⁡(−23)≈−33.69∘ \theta \approx \arctan\left(\frac{-2}{3}\right) \approx -33.69^{\circ} .\newlineSince this angle is in the fourth quadrant, we add 360360 degrees to get the direction angle in the range [00, 360360) degrees.\newlineθ=−33.69∘+360∘ \theta = -33.69^{\circ} + 360^{\circ} .
  5. Find Final Angle: Perform the addition to find the final direction angle:\newlineθ≈−33.69∘+360∘≈326.31∘ \theta \approx -33.69^{\circ} + 360^{\circ} \approx 326.31^{\circ} .

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