Calculate the addition of two or three vectors. This vector addition calculator will determine the resultant vector’s components (x, y, and optional z), magnitude, and direction angle (in the x–y plane).
Vector Addition Formula
Vector addition is performed component-wise: you add the corresponding components of each vector to get the components of the resultant vector. For example:
\begin{aligned}
\vec{R} &= \vec{A} + \vec{B} \;(\text{+ }\vec{C}\text{ if a third vector is included}) \\
R_x &= A_x + B_x \;(\text{+ }C_x) \\
R_y &= A_y + B_y \;(\text{+ }C_y) \\
R_z &= A_z + B_z \;(\text{+ }C_z)\quad \text{(3D)}
\end{aligned}How to add two vectors
The following is a step-by-step guide on adding vectors.
- The first step is to find the coordinates (components) of each vector. For this example, we will assume the first vector is (1,2,3) and the second vector is (4,5,6).
- The next step is to add the corresponding components. For example, the X-component would be 1+4 = 5. The resulting vector is (5,7,9).
- Finally, review the result to make sure each component was added correctly.
Vector Definition
A vector is a quantity with both magnitude and direction. Vectors can be represented in 2D or 3D space (and more generally in higher-dimensional vector spaces), and they are typically drawn as arrows when visualized. They are widely used in physics (for example, in mechanics) to describe quantities such as displacement, velocity, acceleration, and force.
For example, imagine a bowling ball rolling across the ground. The ball has a velocity vector that tells us about its motion; the velocity has both a magnitude (speed) and a direction.
This same bowling ball also has a momentum vector. Momentum is defined as p = m·v, where m is mass (a scalar) and v is velocity (a vector). Because mass is a scalar, it scales the magnitude of the velocity vector and (for positive mass) does not change its direction.
A scalar is a quantity with magnitude only (no direction), such as mass, time, temperature, or energy. Scalars are not inherently “modifiers” of vectors, but a scalar can multiply (scale) a vector, changing its magnitude (and reversing its direction if the scalar is negative).
FAQ
How do you calculate the magnitude of a resultant vector?
To calculate the magnitude of a resultant vector from its components, use the component formula. In 2D: |R| = √(x² + y²). In 3D: |R| = √(x² + y² + z²). If two vectors are perpendicular and you are working with their perpendicular components, this reduces to the Pythagorean theorem.
Can vector addition be used for vectors in three-dimensional space?
Yes, vector addition can be applied in three-dimensional space by adding the corresponding components of the vectors along the x, y, and z axes.
What is the significance of the direction of a vector in physics?
In physics, the direction of a vector is crucial because it indicates where a quantity (such as force or velocity) is pointing. Direction is necessary for correctly predicting how physical systems behave.
