Calculate Vector

Vector Addition Calculator

Enter the magnitude and angle (in degrees) for two vectors to find their resultant vector.

Resultant Vector:

Magnitude:

Angle:

Understanding Vectors and Their Addition

In physics and mathematics, a vector is a quantity that has both magnitude (size) and direction. Unlike scalar quantities (like temperature or mass) which only have magnitude, vectors are used to represent quantities such as force, velocity, displacement, and acceleration. Understanding how to add vectors is fundamental to solving many problems in mechanics, engineering, and even computer graphics.

What is a Vector?

Imagine you're pushing a box. The strength with which you push is the magnitude of the force, and the direction in which you push is the direction of the force. Both are crucial to describe the action completely. This is a vector quantity. Vectors are often represented graphically as arrows, where the length of the arrow indicates its magnitude and the way the arrow points indicates its direction.

Why Add Vectors?

When multiple forces act on an object, or an object undergoes multiple displacements, the net effect is the sum of these individual vectors. For example:

• Forces: If two people push a heavy object from different directions, the object will move in a direction and with a speed that is the resultant of their combined efforts.
• Velocities: A boat moving across a river experiences its own velocity relative to the water, plus the velocity of the river current. Its actual path and speed (resultant velocity) are found by adding these two vectors.
• Displacements: If you walk 5 meters east and then 3 meters north, your final position relative to your starting point is a resultant displacement vector.

How the Calculator Works: The Component Method

Our Vector Addition Calculator uses the component method, which is a robust way to add any number of vectors. Here's how it works:

1. Decomposition: Each vector is broken down into its horizontal (x) and vertical (y) components. If a vector has magnitude 'M' and an angle 'θ' (measured counter-clockwise from the positive x-axis), its components are:
   • X-component (Mx) = M * cos(θ)
   • Y-component (My) = M * sin(θ)
   The calculator automatically converts your input angles from degrees to radians, as trigonometric functions in programming languages typically use radians.
2. Summation: All the x-components are added together to get the resultant x-component (Rx), and all the y-components are added together to get the resultant y-component (Ry).
   • Rx = M1x + M2x + …
   • Ry = M1y + M2y + …
3. Recomposition: The resultant x (Rx) and y (Ry) components are then used to find the magnitude and direction of the resultant vector.
   • Resultant Magnitude (R): R = √(Rx² + Ry²)
   • Resultant Angle (θR): θR = atan2(Ry, Rx)
   The atan2 function is particularly useful as it correctly determines the angle in all four quadrants, providing an angle between -180° and +180°. The calculator then normalizes this to be between 0° and 360°.

Example Calculation

Let's say you have two vectors:

• Vector 1: Magnitude = 10 units, Angle = 0 degrees (purely along the positive x-axis)
• Vector 2: Magnitude = 5 units, Angle = 90 degrees (purely along the positive y-axis)

Using the calculator:

1. Vector 1 Components:
   • X1 = 10 * cos(0°) = 10 * 1 = 10
   • Y1 = 10 * sin(0°) = 10 * 0 = 0
2. Vector 2 Components:
   • X2 = 5 * cos(90°) = 5 * 0 = 0
   • Y2 = 5 * sin(90°) = 5 * 1 = 5
3. Resultant Components:
   • Rx = X1 + X2 = 10 + 0 = 10
   • Ry = Y1 + Y2 = 0 + 5 = 5
4. Resultant Magnitude:
   • R = √(10² + 5²) = √(100 + 25) = √125 ≈ 11.1803 units
5. Resultant Angle:
   • θR = atan2(5, 10) ≈ 0.4636 radians ≈ 26.5651 degrees

The calculator will output a resultant vector with a magnitude of approximately 11.1803 and an angle of approximately 26.5651 degrees. This demonstrates how the component method efficiently combines vectors to find their net effect.