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u=(5,8)\vec{u} = (-5,-8)\newlineFind the direction angle of u\vec{u}. \newlineEnter your answer as an angle in degrees between 00^\circ and 360360^\circ rounded to the nearest hundredth.\newlineθ=\theta = \square^\circ

Full solution

Q. u=(5,8)\vec{u} = (-5,-8)\newlineFind the direction angle of u\vec{u}. \newlineEnter your answer as an angle in degrees between 00^\circ and 360360^\circ rounded to the nearest hundredth.\newlineθ=\theta = \square^\circ
  1. Identify Formula: Identify the formula for the direction angle of a vector. The direction angle θ\theta of a vector u\vec{u} with components (5,8)(-5,-8) can be found using the arctangent function: θ=arctan(y/x)\theta = \text{arctan}(y/x), where xx and yy are the components of the vector. However, since the vector is in the third quadrant, we need to add 180180 degrees to the result of the arctangent to get the angle in the correct range.
  2. Calculate Arctangent: Calculate the arctangent of the y-component divided by the x-component. \newlineθ=arctan(yx)=arctan(85)=arctan(85)\theta = \arctan(\frac{y}{x}) = \arctan(\frac{-8}{-5}) = \arctan(\frac{8}{5})\newlineUsing a calculator, we find that arctan(85)58.00\arctan(\frac{8}{5}) \approx 58.00 degrees.
  3. Adjust for Quadrant: Adjust the angle for the correct quadrant.\newlineSince the vector is in the third quadrant, we add 180180 degrees to the angle found in Step 22.\newlineθ=58.00+180=238.00\theta = 58.00 + 180 = 238.00 degrees
  4. Round Direction Angle: Round the direction angle to the nearest hundredth. \newlineθ238.00\theta \approx 238.00 degrees \newlineHowever, we need to ensure the angle is between 00 and 360360 degrees. Since 238.00238.00 is already within this range, no further adjustments are needed.

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