Find the direction angle of uu=(-10,7) . Enter your answer as an angle in degrees between and rounded to the nearest hundredth.

Definition of Direction Angle: The direction angle of a vector in the plane is the angle measured counterclockwise from the positive x-axis to the vector. To find this angle, we use the arctangent function, which gives us the angle whose tangent is the ratio of the y-coordinate to the x-coordinate of the vector.Calculate Tangent Ratio: For the vector uu=(-10,7), the x-coordinate is -10 and the y-coordinate is 7. The tangent of the direction angle θ is therefore tan(θ) = 7/(-10) = -0.7.Use Arctangent Function: We use the arctangent function to find the angle θ such that tan(θ) = -0.7. However, since the arctangent function returns values between -π/2 and π/2 (or -90 degrees and 90 degrees), and our vector is in the second quadrant (where both sine and cosine are negative), we need to add 180 degrees to the arctangent value to get the correct direction angle.Find Angle in Second Quadrant: Calculating the arctangent of -0.7 using a calculator, we get θ = arctan(-0.7) ≈ -35.00 degrees. Since the vector is in the second quadrant, we add 180 degrees to this value to find the actual direction angle.Calculate Final Direction Angle: Adding 180 degrees to -35.00 degrees, we get the direction angle as 180 - 35.00 = 145.00 degrees.

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Find the direction angle of $\mathbf{u} = (-10,7)$ . Enter your answer as an angle in degrees between $0^\circ$ and $360^\circ$ rounded to the nearest hundredth.

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Q. Find the direction angle of

\mathbf{u} = (-10,7)

. Enter your answer as an angle in degrees between

0^\circ

and

360^\circ

rounded to the nearest hundredth.

Definition of Direction Angle: The direction angle of a vector in the plane is the angle measured counterclockwise from the positive $x$ -axis to the vector. To find this angle, we use the arctangent function, which gives us the angle whose tangent is the ratio of the $y$ -coordinate to the $x$ -coordinate of the vector.
Calculate Tangent Ratio: For the vector $\mathbf{u} = (-10,7)$ , the x-coordinate is $-10$ and the y-coordinate is $7$ . The tangent of the direction angle $\theta$ is therefore $\tan(\theta) = \frac{7}{-10} = -0.7$ .
Use Arctangent Function: We use the arctangent function to find the angle $\theta$ such that $\tan(\theta) = -0.7$ . However, since the arctangent function returns values between $-\pi/2$ and $\pi/2$ (or $-90$ degrees and $90$ degrees), and our vector is in the second quadrant (where both sine and cosine are negative), we need to add $180$ degrees to the arctangent value to get the correct direction angle.
Find Angle in Second Quadrant: Calculating the arctangent of $-0.7$ using a calculator, we get $\theta = \text{arctan}(-0.7) \approx -35.00$ degrees. Since the vector is in the second quadrant, we add $180$ degrees to this value to find the actual direction angle.
Calculate Final Direction Angle: Adding $180$ degrees to $-35.00$ degrees, we get the direction angle as $180 - 35.00 = 145.00$ degrees.