Vector Dot + Cross Product (3D)
a·b = scalar. a×b = vector perpendicular to both.
Result
a · b (dot) / a × b (cross)
32.0000 / (-3.00, 6.00, -3.00)
Angle between: 12.93°.
- a(1, 2, 3)
- b(4, 5, 6)
- |a|3.7417
- |b|8.7750
- a · b32.000000
- a × b(-3.0000, 6.0000, -3.0000)
- cos θ0.974632
- θ12.9332°
Step-by-step
- Dot = aₓbₓ + aᵧbᵧ + a_zb_z = 32.0000.
- Cross_x = aᵧb_z − a_zbᵧ; Cross_y = a_zbₓ − aₓb_z; Cross_z = aₓbᵧ − aᵧbₓ.
How to use this calculator
- Enter components of vectors a + b.
About this calculator
Vector dot product: a·b = aₓbₓ + aᵧbᵧ + a_zb_z. Geometric: |a||b|cos θ. Result is scalar. Used for projections, angle finding, work physics. Cross product: vector perpendicular to both. Right-hand rule. Magnitude |a×b| = |a||b|sin θ = parallelogram area. Used for torque, angular momentum, normal vectors in graphics. Source: NIST DLMF; Wolfram MathWorld - Vector Multiplication.
Frequently asked
Dot: angle, projection, work. Cross: torque, normals, perpendicular construction.
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