Vector Calculator
Enter the components of two vectors to calculate their dot product, cross product, magnitudes, and the angle between them.
Last updated: September 5, 2026
Frequently asked questions
What does the dot product tell you?
A single number summarizing how much two vectors point in the same direction, scaled by their lengths - it's positive if the vectors point roughly the same way, negative if roughly opposite, and zero if they're perpendicular.
What does the cross product tell you?
A new vector perpendicular to both input vectors (only defined in 3D), whose magnitude equals the area of the parallelogram formed by the two vectors - commonly used to find a normal vector to a plane or to calculate torque and angular momentum in physics.
How do I find the angle between two vectors?
cos(θ) = (A·B) / (|A|×|B|), then take the inverse cosine (arccos) of that value to get the angle in degrees.
How do I use this for 2D vectors?
Set both z components to 0 - the dot product and angle calculations work identically, and the cross product will only have a z component, which represents the 2D "cross product" (a scalar in most 2D contexts).
What does it mean if the dot product of two vectors is zero?
The vectors are perpendicular (orthogonal) to each other - regardless of their individual magnitudes, a zero dot product means there's no component of one vector pointing along the other.
What's the magnitude of the vector (4, 5, 6)?
About 8.775 - magnitude is √(x² + y² + z²), so √(4² + 5² + 6²) = √(16+25+36) = √77 ≈ 8.775, calculated for each vector separately by this tool.
Why is the cross product order-dependent (A×B vs B×A)?
A×B and B×A point in exactly opposite directions (same magnitude, flipped sign) because the cross product is anti-commutative - swapping the order reverses which way the 'right-hand rule' points the resulting perpendicular vector.