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vec(u)=(-10,7) 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=(−10,7)\vec{u} = (-10,7)\newlineFind the direction angle of u⃗\vec{u}. Enter your answer as an angle in degrees between \newline0∘0^{\circ} and 360∘360^{\circ} rounded to the nearest hundredth.\newlineθ=□∘\theta=\square^{\circ}

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Q. u⃗=(−10,7)\vec{u} = (-10,7)\newlineFind the direction angle of u⃗\vec{u}. Enter your answer as an angle in degrees between \newline0∘0^{\circ} and 360∘360^{\circ} rounded to the nearest hundredth.\newlineθ=□∘\theta=\square^{\circ}
  1. Identify Formula: Identify the formula for the direction angle of a vector.\newlineThe direction angle θ\theta of a vector u⃗=(x,y)\vec{u}=(x,y) can be found using the arctangent function: θ=arctan(y/x)\theta = \text{arctan}(y/x). However, since arctan\text{arctan} only gives values from −90-90 to 9090 degrees, we need to adjust the angle based on the quadrant in which the vector lies.
  2. Calculate Arctangent: Calculate the arctangent of the y-coordinate divided by the x-coordinate.\newlineFor u⃗=(−10,7)\vec{u}=(-10,7), we have x=−10x=-10 and y=7y=7. Thus, θ=arctan⁡(7−10)=arctan⁡(−0.7)\theta = \arctan\left(\frac{7}{-10}\right) = \arctan(-0.7).\newlineUsing a calculator, we find that arctan⁡(−0.7)≈−35.54\arctan(-0.7) \approx -35.54 degrees.
  3. Adjust Based on Quadrant: Adjust the angle based on the quadrant.\newlineSince the x-coordinate is negative and the y-coordinate is positive, u⃗\vec{u} lies in the second quadrant. In the second quadrant, we must add 180180 degrees to the arctangent value to find the correct direction angle.
  4. Add 180180 Degrees: Add 180180 degrees to the arctangent value.\newlineθ=−35.54\theta = -35.54 degrees +180+ 180 degrees =144.46= 144.46 degrees.
  5. Round to Nearest Hundredth: Round the direction angle to the nearest hundredth. \newlineθ≈144.46\theta \approx 144.46 degrees (rounded to the nearest hundredth).

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