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vec(u)=(4,-2) 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,−2)\vec{u}=(4,-2)\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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Full solution

Q. u⃗=(4,−2)\vec{u}=(4,-2)\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. Identify Components and Formula: Identify the components of vector uu and the formula for the direction angle.\newlineVector uu has components u=(4,−2)u = (4, -2). The direction angle θ\theta of a vector can be found using the arctangent function, specifically θ=arctan(y/x)\theta = \text{arctan}(y/x) where xx and yy are the components of the vector.
  2. Calculate Arctangent: Calculate the arctangent of the y-component divided by the x-component. θ=arctan⁡(−24)=arctan⁡(−0.5)\theta = \arctan(\frac{-2}{4}) = \arctan(-0.5)
  3. Use Calculator for Value: Use a calculator to find the value of θ\theta in radians.θ≈arctan⁡(−0.5)≈−0.463647609\theta \approx \arctan(-0.5) \approx -0.463647609 radians
  4. Convert Radians to Degrees: Convert the angle from radians to degrees. θ≈−0.463647609\theta \approx -0.463647609 radians ×(180π)\times (\frac{180}{\pi}) degrees/radian ≈−26.56505118\approx -26.56505118 degrees
  5. Adjust Negative Angle: Since the angle is negative and we want an angle between 00 and 360360 degrees, add 360360 degrees to the calculated angle.\newlineθ≈−26.56505118\theta \approx -26.56505118 degrees +360+ 360 degrees ≈333.4349488\approx 333.4349488 degrees
  6. Round to Nearest Hundredth: Round the direction angle to the nearest hundredth. θ≈333.43\theta \approx 333.43 degrees

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