Complex Math Calculator

2D Vector Resultant Calculator

This calculator determines the magnitude and direction of the resultant vector when two 2D vectors are added together. Input the magnitude and angle (in degrees relative to the positive X-axis) for each of your two vectors.

Understanding 2D Vector Addition

Vectors are mathematical objects that have both magnitude (size) and direction. In two dimensions, a vector can be represented by its components along the X and Y axes, or by its magnitude and an angle relative to a reference direction (usually the positive X-axis).

How Vector Addition Works

When you add two vectors, you are essentially finding a single vector that represents the combined effect of the original two. This is often visualized using the "parallelogram rule" or "head-to-tail method". Mathematically, vector addition is performed by adding their respective components:

• If Vector 1 is (X1, Y1) and Vector 2 is (X2, Y2), then the Resultant Vector is (X1 + X2, Y1 + Y2).

Converting Magnitude and Angle to Components

To add vectors given in magnitude-angle form, we first need to convert them into their X and Y components. If a vector has magnitude 'M' and angle 'θ' (in degrees) relative to the positive X-axis:

• X-component = M × cos(θ)
• Y-component = M × sin(θ)

It's crucial to convert the angle from degrees to radians before using trigonometric functions in most programming languages (like JavaScript), as these functions typically expect radian input. The conversion is: radians = degrees × (π / 180).

Calculating Resultant Magnitude and Angle

Once you have the resultant X-component (Rx) and Y-component (Ry) by summing the individual components (Rx = X1 + X2, Ry = Y1 + Y2), you can find the resultant vector's magnitude and angle:

• Resultant Magnitude: R = √(Rx² + Ry²) (Pythagorean theorem)
• Resultant Angle: θ_R = atan2(Ry, Rx). The atan2 function is particularly useful as it correctly determines the angle in all four quadrants, returning a value typically between -180° and +180°. This can then be adjusted to a 0° to 360° range if desired.

Example Calculation:

Let's use the default values:

• Vector 1: Magnitude = 10, Angle = 0°
• Vector 2: Magnitude = 5, Angle = 90°

1. Convert Angles to Radians:
   • Angle 1 (rad) = 0° × (π/180) = 0 rad
   • Angle 2 (rad) = 90° × (π/180) = π/2 rad ≈ 1.5708 rad
2. Calculate Components:
   • Vector 1:
     • X1 = 10 × cos(0) = 10 × 1 = 10
     • Y1 = 10 × sin(0) = 10 × 0 = 0
   • Vector 2:
     • X2 = 5 × cos(π/2) = 5 × 0 = 0
     • Y2 = 5 × sin(π/2) = 5 × 1 = 5
3. Sum Components:
   • Resultant X (Rx) = X1 + X2 = 10 + 0 = 10
   • Resultant Y (Ry) = Y1 + Y2 = 0 + 5 = 5
4. Calculate Resultant Magnitude:
   • R = √(10² + 5²) = √(100 + 25) = √125 ≈ 11.18
5. Calculate Resultant Angle:
   • θ_R = atan2(5, 10) ≈ 0.4636 radians
   • θ_R (degrees) = 0.4636 × (180/π) ≈ 26.57°

So, the resultant vector has a magnitude of approximately 11.18 and an angle of approximately 26.57 degrees.

Resultant Magnitude:

Resultant Angle (degrees):

(Resultant X-component: " + resultantX.toFixed(4) + ", Resultant Y-component: " + resultantY.toFixed(4) + ")