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vec(u)=(-9,3) 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=(−9,3) \vec{u}=(-9,3) \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=(−9,3) \vec{u}=(-9,3) \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. Use Arctangent Function: To find the direction angle of the vector u⃗\vec{u} with components (−9,3)(-9,3), we need to use the arctangent function, which gives us the angle whose tangent is the ratio of the y-component to the x-component of the vector. The formula for the direction angle θ\theta is θ=arctan⁡(yx)\theta = \arctan(\frac{y}{x}). However, since the vector is in the second quadrant (x is negative and y is positive), we need to add 180180 degrees to the angle we get from the arctangent function to get the correct direction angle.
  2. Calculate Arctangent: First, calculate the arctangent of the ratio of the y-component to the x-component of the vector u⃗\vec{u}. The y-component is 33 and the x-component is −9-9. So, arctan(3−9)=arctan(−13)\text{arctan}(\frac{3}{-9}) = \text{arctan}(-\frac{1}{3}).
  3. Find Correct Angle: Using a calculator, we find that arctan⁡(−13)\arctan(-\frac{1}{3}) is approximately −18.43-18.43 degrees. Since the arctangent function can return values between −90-90 and 9090 degrees, and our vector is in the second quadrant, we need to add 180180 degrees to this value to find the correct direction angle.
  4. Add 180180 Degrees: Adding 180180 degrees to −18.43-18.43 degrees gives us 161.57161.57 degrees. This is the direction angle of the vector u⃗\vec{u} in the range of 00 to 360360 degrees.

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