The vector F is (12.0, −3.00) in rectangular notation. What is this same vector in polar notation? Express the angle as a positive number between 0° and 360°.
I got 12.4, just can't get the angle.
what is arc tan -3/12
for a positive number, add 360
To convert a vector from rectangular notation to polar notation, you can use the following steps:
Step 1: Find the magnitude (r) of the vector using the formula:
r = √(x^2 + y^2)
Given that F = (12.0, -3.00), the magnitude can be calculated as follows:
r = √((12.0)^2 + (-3.00)^2)
r = √(144 + 9)
r = √153
r ≈ 12.37 (rounded to two decimal places)
Step 2: Find the angle (θ) of the vector using the formula:
θ = atan(y/x)
Given that F = (12.0, -3.00), the angle can be calculated as follows:
θ = atan(-3.00/12.0)
θ ≈ -0.24498 (in radians)
To express the angle as a positive number between 0° and 360°, you need to convert it from radians to degrees, and then adjust it based on the quadrant of the vector.
Step 3: Convert the angle from radians to degrees:
θ_deg = θ * (180/π)
θ_deg ≈ -14.04° (rounded to two decimal places)
Since the angle is negative, you need to adjust it to be within the range of 0° to 360°.
Step 4: Adjust the angle based on the quadrant:
- If x > 0 and y > 0 (quadrant I): no adjustment needed
- If x < 0 and y > 0 (quadrant II): add 180° to the angle
- If x < 0 and y < 0 (quadrant III): add 180° to the angle
- If x > 0 and y < 0 (quadrant IV): add 360° to the angle
In this case, F = (12.0, -3.00) lies in quadrant IV, so you need to add 360° to the angle:
θ_adjusted = θ_deg + 360°
θ_adjusted ≈ 345.96° (rounded to two decimal places)
Therefore, the vector F in polar notation is approximately (12.37, 345.96°).