Vector Calculator

Calculate the dot product, cross product, magnitude, and angle between two vectors instantly, with a live animated diagram showing exactly how they point in space.

What is Vector Calculator?

A vector calculator performs the fundamental operations of vector algebra: the dot product, cross product, magnitude, and the angle between two vectors. Vectors describe quantities that have both magnitude and direction — velocity, force, displacement, and acceleration are all vectors — which makes vector math essential to physics, engineering, 3D graphics, robotics, and machine learning.

The dot product measures how aligned two vectors are and is used to calculate work done by a force or to test perpendicularity. The cross product produces a vector perpendicular to both inputs and is used to find torque, angular momentum, and surface normals in 3D graphics. This calculator computes all of these instantly and draws an animated 2D diagram of your vectors so you can see the geometry, not just the numbers.

Formula

Dot Product: A·B = AxBx + AyBy + AzBz

Cross Product: A×B = (AyBz − AzBy, AzBx − AxBz, AxBy − AyBx)

Magnitude: |A| = √(Ax² + Ay² + Az²)

Angle: θ = arccos( (A·B) / (|A|·|B|) )

How to Calculate

Enter the x, y, and z components of Vector A.
Enter the x, y, and z components of Vector B (leave z at 0 for 2D vectors).
View the dot product, cross product, magnitude of each vector, and the angle between them instantly.
Watch the animated diagram redraw to show how the two vectors point relative to each other.

Example

For A = (3, 2, 0) and B = (1, 4, 0): the dot product is 3×1 + 2×4 = 11, the magnitude of A is √13 ≈ 3.61, the magnitude of B is √17 ≈ 4.12, and the angle between them is about 42°.

Key Benefits

Instant dot product, cross product, magnitude, and angle in one place
Animated diagram shows the geometric relationship, not just numbers
Works for both 2D vectors (z = 0) and full 3D vectors
No manual trigonometry or square-root calculations required

Common Mistakes to Avoid

Confusing the dot product (a scalar) with the cross product (a vector)
Forgetting the cross product only applies to 3D vectors, not 2D
Using degrees vs. radians inconsistently when computing the angle by hand
Assuming a zero cross product means the vectors are unrelated, when it actually means they're parallel

Pro Tips

A dot product of zero instantly tells you the vectors are perpendicular — no need to compute the angle
A cross product of the zero vector tells you the two vectors are parallel (or one is zero)
For 2D problems, just leave z at 0 — the formulas still work correctly
Use the right-hand rule to predict the cross product's direction before checking the calculated result

Key Terms Explained

Scalar: A single number with magnitude but no direction, like the dot product result
Magnitude: The length of a vector, calculated with the Pythagorean theorem
Orthogonal: Another word for perpendicular, used when the dot product is zero
Right-hand rule: A convention for determining the direction of a cross product

Common Use Cases

Calculating work done by a force using the dot product in physics problems
Finding surface normals for lighting calculations in 3D computer graphics
Computing torque and angular momentum in mechanical engineering
Determining whether two vectors are perpendicular or parallel in linear algebra coursework
Robotics and game development calculations involving direction and orientation

Frequently Asked Questions

What's the difference between dot product and cross product?

The dot product returns a single scalar number representing how much two vectors point in the same direction — it's zero when they're perpendicular. The cross product returns a new vector perpendicular to both inputs, with a magnitude equal to the area of the parallelogram they form.

How do you find the angle between two vectors?

Divide the dot product of the two vectors by the product of their magnitudes, then take the inverse cosine (arccos) of that ratio. The result is the angle between them in degrees or radians.

What does it mean when the dot product is zero?

A dot product of zero means the two vectors are perpendicular (orthogonal) to each other — there is no component of one vector in the direction of the other.

Can I use this calculator for 2D vectors?

Yes. Simply leave the z-component at 0 for both vectors and the calculator behaves exactly like a 2D vector calculator, including the diagram, which is always drawn in the x-y plane.

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