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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 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=(−10,7) \vec{u}=(-10,7) \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 Ratio: To find the direction angle of the vector u⃗=(−10,7)\vec{u} = (-10,7), we need to calculate the angle that the 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. Use Arctangent Function: 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=7(−10)=−0.7.\tan(\theta) = \frac{y}{x} = \frac{7}{(-10)} = -0.7.
  3. Determine Quadrant: Next, we use the arctangent function to find the angle θ\theta whose tangent is −0.7-0.7. We must be careful to place the angle in the correct quadrant. Since the x-coordinate is negative and the y-coordinate is positive, u⃗\vec{u} lies in the second quadrant, where the direction angles are between 90∘90^\circ and 180∘180^\circ.θ=atan(−0.7)\theta = \text{atan}(-0.7).
  4. Calculate Direction Angle: Using a calculator, we find that: \newlineθ=atan(−0.7)≈−35.00∘\theta = \text{atan}(-0.7) \approx -35.00^\circ.\newlineHowever, this angle is measured from the positive x-axis in the clockwise direction. To find the direction angle between 0∘0^\circ and 360∘360^\circ, we add 180∘180^\circ to this angle because it is in the second quadrant.\newlineθ=−35.00∘+180∘=145.00∘\theta = -35.00^\circ + 180^\circ = 145.00^\circ.
  5. Round to Nearest Hundredth: We round the direction angle to the nearest hundredth as requested: θ≈145.00∘\theta \approx 145.00^\circ.

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