Determine the direction angle (in degrees) for each vector:
Make sure you’re using degrees instead of radians.
If you use a decimal approximation, you must be accurate to at least 3 decimal places.
⟨6,−5⟩ has direction angle: θ= 140.194 but it says im wrong.
What I did is tan^-1(-5/6)= -39.80557109
I then add 180 to -39.80557109 and i got 140.194.
Sketch ⟨6,−5⟩ and it will be in the fourth quadrant
angle in standard position is tan^-1 (5/6) = 39.816°
so in quad IV it would be 360-39.81 = 320.194°
notice this would be coterminal with your angle of -39.816°
Are you not familiar with the CAST rule?
Tan A = -5/6
A = -39.806o = 39.806o S. of E. = 320.194o CCW from +x-axis.
To determine the direction angle in degrees for the vector ⟨6,−5⟩, you need to use the arctan function, as you correctly did.
The correct calculation is:
tan^-1(-5/6) = -0.792 + π (because tan^-1(-5/6) is in the second quadrant) = 2.349 + 180° = 182.349°
Rounding this value to three decimal places gives 182.349°.
Therefore, the correct direction angle for the vector ⟨6,−5⟩ is 182.349°.
To determine the direction angle (in degrees) for a vector, you can follow these steps:
1. Calculate the inverse tangent of the ratio of the vector's y-component to its x-component using the formula: θ = tan^(-1)(y/x).
In this case, for the vector ⟨6, -5⟩, you correctly calculated tan^(-1)(-5/6) = -39.80557109.
2. Check the quadrant in which the vector lies to adjust the direction angle accordingly:
- In the first quadrant, the direction angle remains as calculated.
- In the second quadrant, add 180 degrees to the direction angle.
- In the third quadrant, add 180 degrees to the direction angle.
- In the fourth quadrant, add 360 degrees to the direction angle.
3. Adjust the direction angle according to the quadrant in which the vector lies:
- Since the vector ⟨6, -5⟩ lies in the second quadrant (x > 0, y < 0), add 180 degrees to the direction angle:
-39.80557109 + 180 = 140.1944289.
Therefore, the correct direction angle (in degrees) for the vector ⟨6, -5⟩ is approximately 140.194 (rounded to 3 decimal places).