Vector Multiplication Calculator
Use this tool to calculate dot product, cross product, Hadamard (component-wise) product, or scalar multiplication for 2D and 3D vectors.
What does “multiplying vectors” mean?
Unlike ordinary numbers, vectors can be multiplied in several different ways, and each one gives a different type of result. In practice, the right multiplication method depends on your goal: finding similarity, finding perpendicular direction, scaling size, or combining components.
This calculator gives you four common options so you can quickly test and compare results without manually expanding formulas each time.
1) Dot product (A · B)
The dot product multiplies matching components and adds them together. The result is a single scalar number. It is useful for measuring alignment between vectors, computing work in physics, and projecting one vector onto another.
2) Cross product (A × B)
In 3D, the cross product returns a new vector perpendicular to both inputs. Its magnitude depends on the area of the parallelogram formed by A and B. In 2D, you commonly compute a scalar z-component (signed area indicator).
3) Hadamard product (A ⊙ B)
This is an element-wise multiplication: each component in A is multiplied by the corresponding component in B. It is common in machine learning, signal processing, and numerical methods.
4) Scalar multiplication (k × A)
This scales a vector by a constant. If k is greater than 1, the vector stretches. If 0 < k < 1, it shrinks. If k is negative, direction reverses.
How to use this multiplying vectors calculator
- Select the multiplication type.
- Choose 2D or 3D mode.
- Enter components of Vector A (and Vector B when needed).
- For scalar multiplication, enter value k.
- Click Calculate to view the result and formula breakdown.
Worked examples
Example A: Dot product
If A = (1, 2, 3) and B = (4, 5, 6), then A · B = 1×4 + 2×5 + 3×6 = 32. Since the output is a scalar, this tells you how strongly the vectors point in similar directions.
Example B: Cross product
Using the same vectors, A × B = (-3, 6, -3). This output is a vector perpendicular to both A and B. Cross products are widely used for torque, angular momentum, and 3D orientation computations.
Example C: Scalar multiplication
If k = 2 and A = (1, 2, 3), then k × A = (2, 4, 6). The direction is unchanged, but magnitude doubles.
Where vector multiplication is used
- Physics: work, force decomposition, torque, angular velocity.
- Computer graphics: lighting, normals, camera orientation, transformations.
- Robotics: motion planning, coordinate frames, kinematics.
- Data science: similarity measures and embedding operations.
- Engineering: stress analysis, electromagnetic field computations.
Common mistakes to avoid
- Mixing 2D and 3D formulas incorrectly.
- Confusing dot product (scalar) with cross product (vector in 3D).
- Dropping signs when computing cross-product components.
- Using non-matching dimensions for element-wise multiplication.
- Forgetting that scalar multiplication does not require Vector B.
Quick FAQ
Is vector multiplication commutative?
Dot and Hadamard products are commutative (A·B = B·A and A⊙B = B⊙A). Cross product is anti-commutative: A×B = −(B×A).
Can I do cross product in 2D?
Yes, as a scalar z-component: AxBy − AyBx. It indicates clockwise/counterclockwise orientation and signed area behavior.
Why does dimension matter?
Because formulas depend on component count. A 3D cross product needs x, y, and z components, while 2D generally uses x and y only.