Vector Dot Product
A·B = aₓbₓ + aᵧbᵧ + a_zb_z (scalar). Also A·B = |A||B|cos θ. Orthogonal when A·B = 0. Used for angles, projections, and work in physics.
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A·B = 0 means perpendicular. cos θ = A·B/(|A||B|). Work = F·d (force dot displacement).
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Why: Dot product gives angle between vectors, work (F·d), and projections. Orthogonality (A·B=0) is fundamental in linear algebra and computer graphics.
How: Component form: sum products of corresponding components. Angle: θ = arccos(A·B/(|A||B|)). Orthogonal when A·B = 0. Projection scalar = A·B̂.
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Enter Two Vectors
Vector A
Vector B
For educational and informational purposes only. Verify with a qualified professional.
🧮 Fascinating Math Facts
A·B = aₓbₓ + aᵧbᵧ + a_zb_z.
— Component Form
cos θ = A·B/(|A||B|).
— Angle
Key Takeaways
- • The dot product is a scalar: .
- • Equivalently: where θ is the angle between the vectors.
- • Orthogonal vectors have dot product 0: A·B = 0 ⟺ A ⊥ B.
- • Projection of A onto B: .
- • Dot product > 0 means acute angle; < 0 means obtuse.
Did You Know?
Work = F·d. The dot product measures how much force acts in the direction of displacement.
Lighting uses dot products: intensity ∝ N·L where N is surface normal, L is light direction.
Cosine similarity is the dot product of normalized vectors. Used in recommendation systems.
The dot product gives the signed length of the projection of one vector onto another.
Orthogonal vectors are "independent" in the sense that one has no component along the other.
A·A = |A|². The dot product of a vector with itself is its squared magnitude.
Understanding the Dot Product
The dot product (scalar product) of two vectors:
When A·B = 0 and neither vector is zero, the vectors are perpendicular (orthogonal).
Expert Tips
Orthogonality
A·B = 0 is the fastest way to check if two vectors are perpendicular.
Angle from Dot
θ = arccos( (A·B) / (|A||B|) ). Always clamp the argument to [-1,1] for numerical stability.
Projection
proj_B(A) = (A·B/|B|) * (B/|B|). The scalar (A·B)/|B| is the signed length.
Sign Interpretation
A·B > 0: acute angle. A·B < 0: obtuse. A·B = 0: right angle.
Frequently Asked Questions
What is the dot product?
A scalar: A·B = a_x b_x + a_y b_y + a_z b_z = |A||B|cos θ.
When is the dot product zero?
When the vectors are perpendicular (orthogonal), or when one is the zero vector.
How do I find the angle between vectors?
Use θ = arccos( (A·B) / (|A||B|) ).
What is the projection of A onto B?
A scalar: (A·B)/|B|. The vector projection is (A·B/|B|²) * B.
Is the dot product commutative?
Yes. A·B = B·A.
What does a negative dot product mean?
The angle between vectors is greater than 90° (obtuse).
How is this used in physics?
Work = F·d. Power = F·v. The dot product measures "how much" one vector acts in the direction of another.
How to Use This Calculator
- Enter components of vectors A and B.
- Click "Calculate" to get dot product, angle, orthogonality, and projections.
- Review the visualization showing the angle between vectors.
- Copy results for homework or reports.
Disclaimer: This calculator uses standard floating-point arithmetic. For zero vectors, the angle is undefined. Results are suitable for educational and professional use.
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