vec(u)=(-10,-9) 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=◻" 。 "

Question

vec(u)=(-10,-9) 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=◻" 。 "

Bytelearn · Accepted Answer

Calculate Ratio Arctangent: To find the direction angle of the vector $ \vec{u} = (-10, -9) $, we need to calculate the angle that this vector makes with the positive x-axis. The direction angle $ \theta $ can be found using the arctangent function (also known as the inverse tangent or atan), which gives us the angle whose tangent is the ratio of the y-coordinate to the x-coordinate of the vector.Convert Radians to Degrees: First, we calculate the arctangent of the ratio of the y-coordinate to the x-coordinate of the vector $ \vec{u} $. Since $ \vec{u} = (-10, -9) $, the ratio is $ \frac{-9}{-10} $. Using a calculator, we find that $ \arctan\left(\frac{-9}{-10}\right) $ gives us an angle in radians.Adjust for Quadrant: We convert the angle from radians to degrees because the question asks for the answer in degrees. To convert radians to degrees, we multiply by $ \frac{180}{\pi} $.Perform Calculation: The calculated angle will give us the angle relative to the positive x-axis, but since both the x and y coordinates of $ \vec{u} $ are negative, $ \vec{u} $ lies in the third quadrant. The arctangent function will give us an angle in the first quadrant, so we need to add 180° to get the correct direction angle in the third quadrant.Add 180 Degrees: Performing the calculation, we have $ \theta = \arctan\left(\frac{-9}{-10}\right) \times \frac{180}{\pi} + 180° $. Using a calculator, we find that $ \theta \approx 42.01° + 180° $.Round to Nearest Hundredth: Adding 180° to 42.01°, we get $ \theta \approx 222.01° $. This is the direction angle of vector $ \vec{u} $ in the third quadrant, measured counterclockwise from the positive x-axis.We round the direction angle to the nearest hundredth as the question asks. Therefore, the direction angle of vector $ \vec{u} $ is approximately 222.01°.

$\vec{u}=(-10,-9)$ $\newline$ Find the direction angle of $\vec{u}$ . Enter your answer as an angle in degrees between $0^{\circ}$ and $360^{\circ}$ rounded to the nearest hundredth. $\newline$ $\theta=\square^{\circ}$

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More problems from Inverses of sin, cos, and tan: degrees

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u⃗=(−10,−9) \vec{u}=(-10,-9) u=(−10,−9)\newlineFind the direction angle of u⃗ \vec{u} u. Enter your answer as an angle in degrees between 0∘ 0^{\circ} 0∘ and 360∘ 360^{\circ} 360∘ rounded to the nearest hundredth.\newlineθ=□∘ \theta=\square^{\circ} θ=□∘

$\vec{u}=(-10,-9)$ $\newline$ Find the direction angle of $\vec{u}$ . Enter your answer as an angle in degrees between $0^{\circ}$ and $360^{\circ}$ rounded to the nearest hundredth. $\newline$ $\theta=\square^{\circ}$