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vec(u)=(6,-8) 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=(6,−8) \vec{u}=(6,-8) \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=(6,−8) \vec{u}=(6,-8) \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=(6,−8)\vec{u} = (6, -8), 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}) 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=−86=−43.\tan(\theta) = \frac{y}{x} = \frac{-8}{6} = \frac{-4}{3}.
  3. Determine Quadrant: Next, we use the arctangent function to find the angle θ\theta whose tangent is −43-\frac{4}{3}. We must be careful to place the angle in the correct quadrant. Since the xx-coordinate is positive and the yy-coordinate is negative, u⃗\vec{u} is in the fourth quadrant where the direction angles are between 270∘270^\circ and 360∘360^\circ.\newlineθ=arctan⁡(−43)\theta = \arctan\left(-\frac{4}{3}\right).
  4. Adjust for Quadrant: Using a calculator, we find the arctangent of −43-\frac{4}{3}. However, this will give us an angle in the second quadrant, so we must add 360∘360^\circ to get the angle in the fourth quadrant.\newlineθ=arctan⁡(−43)+360∘\theta = \arctan(-\frac{4}{3}) + 360^\circ.
  5. Final Calculation: After calculating, we get: θ≈arctan⁡(−43)+360∘≈−53.13∘+360∘≈306.87∘\theta \approx \arctan(-\frac{4}{3}) + 360^\circ \approx -53.13^\circ + 360^\circ \approx 306.87^\circ. We round this to the nearest hundredth as requested.

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