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vec(u)=(6,7) 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=(6,7)\vec{u}=(6,7)\newlineFind the direction angle of \newlineu⃗\vec{u}. Enter your answer as an angle in degrees between \newline0∘0^{\circ} and \newline360∘360^{\circ} rounded to the nearest hundredth.\newlineθ=□∘\theta=\square^{\circ}

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Full solution

Q. u⃗=(6,7)\vec{u}=(6,7)\newlineFind the direction angle of \newlineu⃗\vec{u}. Enter your answer as an angle in degrees between \newline0∘0^{\circ} and \newline360∘360^{\circ} rounded to the nearest hundredth.\newlineθ=□∘\theta=\square^{\circ}
  1. Calculate Tangent Ratio: To find the direction angle of the vector u⃗=(6,7) \vec{u} = (6,7) , we need to calculate the angle θ \theta that the vector makes with the positive x-axis. The direction angle can be found using the arctangent function, which is the inverse of the tangent function. The tangent of the angle is the ratio of the y-coordinate to the x-coordinate of the vector.
  2. Find Arctangent: First, we calculate the tangent of the angle θ \theta using the coordinates of the vector u⃗ \vec{u} :\newlinetan⁡(θ)=yx=76 \tan(\theta) = \frac{y}{x} = \frac{7}{6}
  3. Calculate Angle in Degrees: Next, we find the angle θ \theta by taking the arctangent of 76 \frac{7}{6} :\newlineθ=arctan⁡(76) \theta = \arctan\left(\frac{7}{6}\right) \newlineWe will use a calculator to find the value of θ \theta in degrees.
  4. Check Quadrant: Using a calculator, we find that:\newlineθ≈arctan⁡(76)≈49.3987 degrees \theta \approx \arctan\left(\frac{7}{6}\right) \approx 49.3987 \text{ degrees}
  5. Round to Nearest Hundredth: Since the vector u⃗ \vec{u} is in the first quadrant (both x and y are positive), the direction angle θ \theta is already between 00 and 360360 degrees. Therefore, we do not need to adjust the angle further.
  6. Round to Nearest Hundredth: Since the vector u⃗ \vec{u} is in the first quadrant (both x and y are positive), the direction angle θ \theta is already between 00 and 360360 degrees. Therefore, we do not need to adjust the angle further.Finally, we round the angle to the nearest hundredth as requested:\newlineθ≈49.40∘ \theta \approx 49.40^{\circ}

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