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vec(u)=(-1,4) 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=(−1,4) \vec{u}=(-1,4) \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=(−1,4) \vec{u}=(-1,4) \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 Tangent: To find the direction angle of the vector u⃗=(−1,4)\vec{u} = (-1,4), 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} or atan\text{atan}), which gives us the angle whose tangent is the ratio of the y-coordinate to the x-coordinate of the vector.
  2. Consider Quadrant: First, we calculate the tangent of the angle θ\theta using the coordinates of u⃗\vec{u}. The tangent of θ\theta is the ratio of the y-coordinate to the x-coordinate.\newlinetan⁡(θ)=yx=4(−1)=−4\tan(\theta) = \frac{y}{x} = \frac{4}{(-1)} = -4
  3. Use Arctangent Function: Next, we use the arctangent function to find the angle θ\theta whose tangent is −4-4. However, we must be careful with the signs and the quadrant in which the vector lies. Since the x-coordinate is negative and the y-coordinate is positive, u⃗\vec{u} lies in the second quadrant. The arctangent function will give us an angle in the fourth quadrant, so we need to add 180∘180^\circ to get the angle in the second quadrant.\newlineθ=atan(−4)+180∘\theta = \text{atan}(-4) + 180^\circ
  4. Add 180180°: We calculate the arctangent of −4-4 using a calculator and then add 180°180° to find the direction angle in the second quadrant.\newlineθ≈atan(−4)+180°≈−75.96°+180°≈104.04°\theta \approx \text{atan}(-4) + 180° \approx -75.96° + 180° \approx 104.04°
  5. Round to Nearest Hundredth: We round the direction angle to the nearest hundredth as requested. \newlineθ≈104.04∘\theta \approx 104.04^\circ (rounded to the nearest hundredth)

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