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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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Inverses of sin, cos, and tan: degreesright chevron icon

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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. Definition of direction angle: The direction angle of a vector in the coordinate plane is the angle the vector makes with the positive x-axis. To find this angle, we use the arctangent function (tan⁡−1\tan^{-1}), which gives us the angle whose tangent is the ratio of the y-coordinate to the x-coordinate of the vector.\newlineCalculation: θ=tan⁡−1(yx)=tan⁡−1(−24)=tan⁡−1(−0.5)\theta = \tan^{-1}(\frac{y}{x}) = \tan^{-1}(\frac{-2}{4}) = \tan^{-1}(-0.5).
  2. Calculation of theta: Using a calculator to find the arctangent of −0.5-0.5, we get an initial angle. However, since the vector is in the fourth quadrant (because the xx-coordinate is positive and the yy-coordinate is negative), we need to add 360360 degrees to the initial angle if it's negative to ensure the angle is between 00 and 360360 degrees.\newlineCalculation: Initial angle = tan⁡−1(−0.5)≈−26.57\tan^{-1}(-0.5) \approx -26.57 degrees. Since it's negative, we add 360360 degrees to find the direction angle in the correct range.\newlineTheta = −26.57+360≈333.43-26.57 + 360 \approx 333.43 degrees.

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