Enter two 3D vectors and instantly get the dot product, cross product and magnitude of each.
Vector Calculator
How It’s Calculated
The dot product A·B = a₁b₁ + a₂b₂ + a₃b₃ gives a single number — useful for finding angles between vectors and checking perpendicularity (A·B = 0 means the vectors are perpendicular). The cross product A×B gives a new vector, perpendicular to both A and B, computed component-by-component using the determinant-style formula. Magnitude |A| = √(a₁²+a₂²+a₃²) is the vector’s length.
Worked Example
For A = (1,2,3) and B = (4,5,6): A·B = 1×4 + 2×5 + 3×6 = 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.
Frequently Asked Questions
Q: What does it mean if the dot product is 0?
The two vectors are perpendicular (at 90° to each other) — this is one of the fastest ways to check perpendicularity without computing an angle directly.
Q: Is the cross product a number or a vector?
A vector — unlike the dot product, which gives a scalar. The cross product’s direction is perpendicular to both input vectors, following the right-hand rule.
Dot and cross products are core to the 2nd PUC Vector Algebra chapter, and resurface again in VTU Vector Calculus. See how Daniel Classes teaches 1st and 2nd PUC Mathematics →