Vector Multiplication Calculator

Enter Vector Values

Scaled vectors: A' = scalarA × A, B' = scalarB × B

Dot product: A · B = AxBx + AyBy + AzBz

Cross product: A × B = [AyBz - AzBy, AzBx - AxBz, AxBy - AyBx]

Component product: A ⊙ B = [AxBx, AyBy, AzBz]

Magnitude: |A| = √(Ax² + Ay² + Az²)

Angle: θ = cos⁻¹((A · B) / (|A||B|))

Projection: projB(A) = ((A · B) / |B|²) × B

Scalar triple product: A · (B × C)

1. Enter the X, Y, and Z components for Vector A.
2. Enter the X, Y, and Z components for Vector B.
3. Add Vector C if you need scalar triple product volume.
4. Use scalar multipliers when vectors must be scaled first.
5. Choose decimal precision and enter a unit label.
6. Press the calculate button to show results above the form.
7. Review the chart, table, formulas, and export buttons.

Vector Multiplication Basics

Vector multiplication helps describe direction, size, and interaction between quantities. It is used in geometry, physics, graphics, robotics, navigation, and engineering. A vector has components. Each component shows movement along an axis. This calculator accepts two main vectors. It also accepts a third vector for scalar triple product checks.

Dot Product Meaning

The dot product returns one number. It measures how strongly two vectors point in the same direction. A positive value means both vectors mostly agree. A negative value means they mostly oppose each other. A zero value shows perpendicular direction, when neither vector supports the other along the same line.

Cross Product Meaning

The cross product returns a new vector. This new vector is perpendicular to the first two vectors. Its length equals the parallelogram area formed by both vectors. This is useful for torque, normal vectors, rotation, and surface direction. The calculator also shows triangle area, because it is half the cross product magnitude.

Component Multiplication

Component multiplication multiplies matching coordinates. It is also called the Hadamard product. It is helpful in data work, scaling models, color operations, and quick coordinate checks. It is not the same as dot product or cross product, because it returns another vector.

Scaling and Projection

Scalar multipliers change vector length before calculations. Direction stays the same when the scalar is positive. Direction reverses when the scalar is negative. Projection shows how much one vector falls along another vector. It is useful for shadows, forces, ramps, and vector resolution.

Why This Tool Helps

Manual vector work can become slow. Sign errors are common in cross product steps. Angle conversion also causes mistakes. This tool shows magnitudes, unit vectors, dot product, cross product, projections, areas, and triple product results in one place. The graph gives a fast visual check. CSV export supports spreadsheets. PDF export supports reports and homework notes.

Use Results Carefully

Vector results depend on units. Keep all components in the same unit system. Check whether your problem needs two dimensional or three dimensional logic. Use decimal precision based on your task. For formal work, include formulas, units, and a short note explaining what each vector represents for cleaner final notes.