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Find the direction angle of u=(−7,−10)\mathbf{u}=(-7,-10). Enter your answer as an angle in degrees between 0∘0^\circ and 360∘360^\circ rounded to the nearest hundredth.

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Full solution

Q. Find the direction angle of u=(−7,−10)\mathbf{u}=(-7,-10). Enter your answer as an angle in degrees between 0∘0^\circ and 360∘360^\circ rounded to the nearest hundredth.
  1. Understand Direction Angle: Understand the concept of a direction angle.\newlineThe direction angle of a vector in the plane is the angle measured counterclockwise from the positive xx-axis to the vector. For a vector u=(x,y)\mathbf{u} = (x, y), the direction angle θ\theta can be found using the arctangent function, where θ=arctan⁡(yx)\theta = \arctan(\frac{y}{x}). However, since the arctangent function only gives values from −π2-\frac{\pi}{2} to π2\frac{\pi}{2}, we need to adjust the angle depending on the quadrant in which the vector lies.
  2. Identify Vector Quadrant: Identify the quadrant in which the vector lies.\newlineThe vector u=(−7,−10)\mathbf{u} = (-7, -10) lies in the third quadrant because both xx and yy are negative. In the third quadrant, the direction angle is between 180∘180^\circ and 270∘270^\circ.
  3. Calculate Arctangent: Calculate the arctangent of the vector's y-coordinate divided by its x-coordinate. θ=arctan⁡(yx)=arctan⁡(−10−7)=arctan⁡(107)\theta = \arctan(\frac{y}{x}) = \arctan(\frac{-10}{-7}) = \arctan(\frac{10}{7}). We use a calculator to find the arctangent of 107\frac{10}{7}.
  4. Adjust for Third Quadrant: Adjust the angle for the third quadrant.\newlineSince the arctangent function gives us an angle in the first quadrant, we need to add 180∘180^\circ to get the direction angle in the third quadrant.\newlineθ=arctan⁡(107)+180∘\theta = \arctan(\frac{10}{7}) + 180^\circ.
  5. Calculate Exact Direction Angle: Calculate the exact value of the direction angle.\newlineUsing a calculator, we find that arctan⁡(107)≈55.00∘\arctan(\frac{10}{7}) \approx 55.00^\circ (rounded to two decimal places).\newlineTherefore, θ≈55.00∘+180∘=235.00∘\theta \approx 55.00^\circ + 180^\circ = 235.00^\circ.

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