Polar Form Addition Calculator

Polar Addition Method:

\[ \text{Convert to rectangular: } x = r \cos(\theta),\ y = r \sin(\theta) \] \[ \text{Add components: } x_{sum} = x_1 + x_2,\ y_{sum} = y_1 + y_2 \] \[ \text{Convert back to polar: } r = \sqrt{x_{sum}^2 + y_{sum}^2},\ \theta = \text{atan2}(y_{sum}, x_{sum}) \]

First Vector

Magnitude (r₁):

Angle (θ₁):

radians

Second Vector

Magnitude (r₂):

Angle (θ₂):

radians

Resultant Vector:

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1. What is Polar Form Addition?

Polar form addition involves converting polar coordinates (magnitude and angle) to rectangular form (x and y components), performing vector addition in rectangular form, and then converting the result back to polar coordinates.

2. How the Calculator Works

The calculator performs the following steps:

\[ \begin{aligned} &1.\ \text{Convert to rectangular:} \\ &\quad x_1 = r_1 \cos(\theta_1),\ y_1 = r_1 \sin(\theta_1) \\ &\quad x_2 = r_2 \cos(\theta_2),\ y_2 = r_2 \sin(\theta_2) \\ &2.\ \text{Add components:} \\ &\quad x_{sum} = x_1 + x_2,\ y_{sum} = y_1 + y_2 \\ &3.\ \text{Convert back to polar:} \\ &\quad r = \sqrt{x_{sum}^2 + y_{sum}^2},\ \theta = \text{atan2}(y_{sum}, x_{sum}) \end{aligned} \]

Where:

\( r \) — Magnitude of the vector
\( \theta \) — Angle in radians
\( \text{atan2} \) — Two-argument arctangent function that preserves quadrant information

3. Importance of Polar Addition

Applications: Polar form addition is essential in physics, engineering, and signal processing where vectors are naturally represented in polar form (e.g., phasors in AC circuit analysis).

4. Using the Calculator

Tips: Enter magnitudes as positive numbers and angles in radians. For degrees, convert first (radians = degrees × π/180).

5. Frequently Asked Questions (FAQ)

Q1: Why convert to rectangular form for addition?
A: Vector addition is simpler in rectangular form (component-wise addition) compared to polar form.

Q2: Can I input angles in degrees?
A: The calculator requires radians. Multiply degrees by π/180 to convert (e.g., 90° = π/2 radians).

Q3: What's the range for the resulting angle θ?
A: The result is in the range (-π, π] radians (-180° to 180°).

Q4: What if both vectors have zero magnitude?
A: The result will be 0 ∠ 0. The angle is technically undefined but reported as 0.

Q5: How precise are the results?
A: Results are rounded to 4 decimal places. Intermediate calculations use full precision.