What is the difference between dot product and cross product?

The dot product produces a scalar representing how parallel two vectors are. The cross product produces a new vector perpendicular to both input vectors, with magnitude equal to the parallelogram area they span.

When is the dot product zero?

The dot product is zero when two vectors are perpendicular (orthogonal). This is used extensively in physics to test if forces are at right angles.

What is the cross product used for?

The cross product is used to find a vector normal to a surface, compute torque in mechanics, determine angular momentum, and calculate the area of parallelograms.

Can I use 2D vectors?

Yes — just enter 0 for the k (z) component. The cross product of 2D vectors gives a vector in the z-direction only.

What does vector magnitude represent?

The magnitude (length) of a vector represents the size of the quantity it describes — for example, speed (magnitude of velocity) or force strength (magnitude of a force vector).

Math

↵ to calculate

Vector Calculator - Add Subtract Dot Product Online

Add, subtract, find dot product, cross product, and magnitudes of two 3D vectors simultaneously.

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How to use this calculator

👉 Fill in the boxes below and your answer appears instantly — no maths needed, we do it all for you! 🎉

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In plain English — what does this do?

🔢 This tool does the maths for you! Just type in your numbers, and it gives you the answer right away. No need to count on your fingers or use a pen and paper.

📌 Direct Answer & Summary

A vector calculator performs common operations on two three-dimensional vectors: addition, subtraction, dot product, cross product, and individual magnitudes. Vectors are fundamental in physics, engineering, computer graphics, and machine learning — any situation involving quantities with both magnitude and direction.

Vector A

Vector B

What is Vector Calculator - Add Subtract Dot Product Online?

How to use it

1️⃣ Enter the i, j, k components for vector A.
2️⃣ Enter the i, j, k components for vector B.
3️⃣ All operations (sum, difference, dot product, cross product, magnitudes) are computed and displayed simultaneously.

Formula

A+B = (a₁+b₁, a₂+b₂, a₃+b₃). A−B = (a₁−b₁, a₂−b₂, a₃−b₃). A·B = a₁b₁+a₂b₂+a₃b₃. A×B = (a₂b₃−a₃b₂, a₃b₁−a₁b₃, a₁b₂−a₂b₁). |A| = √(a₁²+a₂²+a₃²).

💡 See it in action — a real example

A = (1,2,3), B = (4,5,6): A+B = (5,7,9). A−B = (−3,−3,−3). A·B = 4+10+18 = 32. A×B = (2×6−3×5, 3×4−1×6, 1×5−2×4) = (−3, 6, −3). |A| = √14 ≈ 3.742.

❓ Common questions

What is the difference between dot product and cross product?: The dot product produces a scalar representing how parallel two vectors are. The cross product produces a new vector perpendicular to both input vectors, with magnitude equal to the parallelogram area they span.
When is the dot product zero?: The dot product is zero when two vectors are perpendicular (orthogonal). This is used extensively in physics to test if forces are at right angles.
What is the cross product used for?: The cross product is used to find a vector normal to a surface, compute torque in mechanics, determine angular momentum, and calculate the area of parallelograms.
Can I use 2D vectors?: Yes — just enter 0 for the k (z) component. The cross product of 2D vectors gives a vector in the z-direction only.
What does vector magnitude represent?: The magnitude (length) of a vector represents the size of the quantity it describes — for example, speed (magnitude of velocity) or force strength (magnitude of a force vector).

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