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vec(u)=(2,5) 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=(2,5) \vec{u}=(2,5) \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=(2,5) \vec{u}=(2,5) \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. The direction angle, often denoted as θ\theta, can be found using the arctangent function (also known as the inverse tangent or atan\text{atan}), which is the ratio of the y-coordinate to the x-coordinate of the vector.
  2. Identifying Vector Components: First, we identify the xx and yy components of the vector u⃗=(2,5)\vec{u} = (2,5). Here, x=2x = 2 and y=5y = 5.
  3. Calculating Direction Angle: Next, we calculate the direction angle using the arctangent function: θ=atan(y/x)=atan(5/2)\theta = \text{atan}(y/x) = \text{atan}(5/2).
  4. Using Arctangent Function: Using a calculator, we find the arctangent of 52\frac{5}{2}: θ≈atan(2.5)≈68.19859\theta \approx \text{atan}(2.5) \approx 68.19859 degrees.
  5. Rounding to Nearest Hundredth: Since the vector is in the first quadrant (where both xx and yy are positive), the direction angle θ\theta is already between 0∘0^\circ and 90∘90^\circ. Therefore, we do not need to adjust the angle, and we can round it to the nearest hundredth:\newlineθ≈68.20∘\theta \approx 68.20^\circ.

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