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Math • Vector Dot & Cross Product Calculator

Vector Dot & Cross Product Calculator

Calculate the dot product, cross product, magnitudes, and angle between two 3D vectors.

Vector u = (u₁, u₂, u₃)
Magnitude |u| = 5.385
Vector v = (v₁, v₂, v₃)
Magnitude |v| = 5.916
Dot Product (u · v)
-19.000
Scalar projection result
Cross Product (u × v)
(-14, 13, 17)
Perpendicular normal vector
Angle Between Vectors (θ)
126.61°
arccos((u·v) / (|u||v|))
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Calculation Guide & Reference

Vector Dot Product, Cross Product & Angle Mathematics

Vector calculator computes the dot product, cross product, magnitudes, and angle between two 3D vectors.

Standardized Mathematical Formula
A·B = AₓBₓ + AᵧBᵧ + A_zB_z | cos(θ) = (A·B) / (|A||B|)

The dot product multiplies corresponding components and sums them, producing a single scalar related to how aligned the two vectors are. Dividing by both magnitudes normalizes this into the cosine of the angle between them.

Variables:
A·B:Dot product — a scalar value
A×B:Cross product — a new vector perpendicular to both A and B
|A|, |B|:Magnitudes (lengths) of vectors A and B
How It Works (Step-by-Step)
  • 1Enter the x, y, z components of two 3D vectors.
  • 2View the dot product, cross product vector, both magnitudes, and the angle between the vectors.
Real-World Numerical Example
A = (3, −2, 4) and B = (1, 5, −3)

Find the dot product, cross product, and angle between vectors A = (3, −2, 4) and B = (1, 5, −3).

•Dot product: (3×1) + (−2×5) + (4×−3) = 3 − 10 − 12 = −19.
•|A| = √(9+4+16) = √29 ≈ 5.385. |B| = √(1+25+9) = √35 ≈ 5.916.
•cos(θ) = −19 ÷ (5.385 × 5.916) ≈ −0.596.
•θ = acos(−0.596) ≈ 126.6°.
Result: The dot product is −19, and the angle between the two vectors is approximately 126.6°.
Calculation Best Practices & Tips
Examine the discriminant (b² - 4ac) before solving quadratics to instantly know if roots are real, distinct, repeated, or complex.
In statistical analysis, always check both the mean and median to identify if data distribution is skewed by outliers.
When calculating percentage changes, ensure you divide by the original starting baseline value: ((New - Old) / Old) * 100.

Frequently Asked Questions (FAQ)

A dot product of zero means the two vectors are perpendicular (orthogonal) — the angle between them is exactly 90°.

The dot product returns a single scalar number measuring alignment; the cross product returns a new vector perpendicular to both inputs, with a magnitude related to the area of the parallelogram they form. The cross product only exists in 3D (and 7D).

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