What Is a Vector Calculator?
This vector calculator works with two-dimensional (x, y) or three-dimensional (x, y, z) vectors and instantly computes the four measurements most often needed in linear algebra, physics, and computer graphics: the dot product, the cross product, the magnitude (length) of each vector, and the angle between them. Enter the components of vector A and vector B, choose 2D or 3D, and every result updates automatically.
How to Use the Vector Calculator
- Choose whether your vectors are 2D (x, y) or 3D (x, y, z).
- Enter the components of vector A and vector B.
- Instantly view the dot product, cross product, magnitude of each vector, and the angle between them.
What Is the Dot Product?
The dot product (also called the scalar product) multiplies two vectors together and returns a single number, not a vector. It is calculated by multiplying corresponding components and adding the results. The dot product tells you how much two vectors point in the same direction: a large positive value means the vectors point roughly the same way, a value near zero means they are roughly perpendicular, and a negative value means they point in largely opposite directions. In physics, the dot product is used to calculate work done by a force (work = force · displacement) and to project one vector onto another. In computer graphics, it is used constantly for lighting calculations, determining the angle between surfaces and light sources, and testing whether two directions are aligned.
What Is the Cross Product?
The cross product is only defined for 3D vectors (a 2D "cross product" is sometimes used informally as a scalar to test rotational direction, but it does not produce a true perpendicular vector in the plane). Unlike the dot product, the cross product of two 3D vectors returns a brand-new vector that is perpendicular to both original vectors, following the right-hand rule. Its magnitude equals the area of the parallelogram formed by the two vectors. The cross product is essential in physics for calculating torque (torque = r × F) and angular momentum, and in 3D computer graphics for finding surface normals — the direction a triangle or polygon "faces," which is critical for lighting, shading, and backface culling.
What Is Vector Magnitude?
The magnitude of a vector (written |a|) is its length — a single non-negative number found using the Pythagorean theorem extended to two or three dimensions. Magnitude is used any time you need to know "how far" or "how fast" without caring about direction: the speed of a moving object (the magnitude of its velocity vector), the distance between two points, or the strength of a force. Magnitude is also required to calculate the angle between two vectors and to normalize a vector (scale it down to a length of exactly 1, which is common in computer graphics for direction vectors).
Angle Between Two Vectors
Combining the dot product and magnitude formulas gives the angle θ between two vectors. This is widely used in engineering to check whether two forces are aligned, in graphics to test how directly a surface faces a light source, and in navigation and robotics to compare headings.
Worked Example
Dot product: a · b = (1×4) + (2×5) + (3×6) = 4 + 10 + 18 = 32
Cross product: a × b = (2×6 − 3×5, 3×4 − 1×6, 1×5 − 2×4) = (−3, 6, −3)
Magnitudes: |a| = √14 ≈ 3.7417, |b| = √77 ≈ 8.7750
Angle: θ = cos⁻¹(32 ÷ (3.7417 × 8.7750)) = cos⁻¹(0.9746) ≈ 12.93°
Common Uses for Vectors
- Physics: force, velocity, acceleration, work, and torque calculations.
- Computer graphics and game development: lighting, surface normals, and camera direction.
- Engineering: analyzing combined forces and structural loads.
- Navigation and robotics: comparing headings and computing turning angles.
Common Mistakes to Avoid
- Trying to compute a true 3D-style cross product for 2D vectors — 2D vectors don't have a perpendicular-vector cross product, only an optional signed scalar.
- Forgetting that the dot product returns a number (scalar), while the cross product returns a vector.
- Mixing up radians and degrees when reporting the angle between vectors.
Frequently Asked Questions
The dot product multiplies two vectors and returns a single number (scalar) describing how aligned they are. The cross product (defined only in 3D) returns a new vector that is perpendicular to both original vectors.
A dot product of zero means the two vectors are perpendicular (orthogonal) to each other — the angle between them is exactly 90°.
Not in the traditional sense. The true cross product (a vector perpendicular to both inputs) only exists in 3D. For 2D vectors, a related scalar value (a₁b₂ − a₂b₁) is sometimes used to test rotational direction or compute area, but it isn't a vector.