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vec(u)=(-4,-3) 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=(−4,−3) \vec{u}=(-4,-3) \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=(−4,−3) \vec{u}=(-4,-3) \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 Slope: To find the direction angle of the vector u⃗=(−4,−3)\vec{u} = (-4, -3), we need to calculate the angle that this vector makes with the positive x-axis. The direction angle, often denoted as θ\theta, can be found using the arctangent function (tan⁡−1\tan^{-1}) of the slope of the vector, which is the ratio of its y-component to its x-component. The formula to find the direction angle is θ=tan⁡−1(yx)\theta = \tan^{-1}(\frac{y}{x}).
  2. Use Arctangent Function: First, we calculate the slope of the vector by dividing its yy-component by its xx-component. For u⃗=(−4,−3)\vec{u} = (-4, -3), the slope is (−3)/(−4)=3/4(-3)/(-4) = 3/4.
  3. Adjust for Quadrant: Next, we use the arctangent function to find the angle that corresponds to this slope. However, since the vector is in the third quadrant (both xx and yy components are negative), the angle given by the arctangent function will be in the fourth quadrant. We need to add 180∘180^\circ to this angle to get the correct direction angle in the third quadrant.\newlineθ=tan⁡−1(34)+180∘\theta = \tan^{-1}(\frac{3}{4}) + 180^\circ
  4. Calculate Arctangent: Now, we calculate the arctangent of 34\frac{3}{4} using a calculator.\newlineθ=tan⁡−1(34)≈36.87∘\theta = \tan^{-1}(\frac{3}{4}) \approx 36.87^\circ
  5. Add 180180°: Finally, we add 180180° to the angle we found to get the direction angle in the correct quadrant.\newlineθ=36.87°+180°\theta = 36.87° + 180°\newlineθ≈216.87°\theta \approx 216.87°

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