doomath Calculator
3D Vector Operations & Cross Product
Compute 3D vector magnitudes, dot product u·v, cross product vector u×v, angle between vectors in degrees & radians, and orthogonal vector projections.
InputsChange a value to recalculate instantly
ResultUpdates as you type
FORMULA & STEPS3D Vector Operations & Cross Product — Mathematical Derivation & Steps
3 StepsSequential Calculation Breakdown
1
Magnitudes
Use the Euclidean norm of each vector.
2
Dot product
Multiply matching components and add them.
3
Angle
Use the normalized dot product.
λ DooMathWork with this calculation
Local · deterministic · optionalQuick mathNatural-language-style shortcuts for deterministic arithmetic.
Calculator guide
How to use the 3D Vector Operations & Cross Product
Compute 3D vector magnitudes, dot product u·v, cross product vector u×v, angle between vectors in degrees & radians, and orthogonal vector projections.
01Set inputsChoose values, units and assumptions.
02Apply the modelu·v = u_x v_x + u_y v_y + u_z v_z; u×v = (u_y v_z - u_z v_y, u_z v_x - u_x v_z, u_x v_y - u_y v_x); cos(θ) = (u·v)/(|u||v|)
03Read the resultCheck the output against the assumptions before using it.
What this calculator does
Specify the components (x, y, z) for vectors u and v. The calculator determines magnitudes, scalar dot product, right-hand rule cross product, and enclosed angle.
Formula & method
u·v = u_x v_x + u_y v_y + u_z v_z; u×v = (u_y v_z - u_z v_y, u_z v_x - u_x v_z, u_x v_y - u_y v_x); cos(θ) = (u·v)/(|u||v|)
Assumptions & notes
- Two nonzero vectors are perpendicular if and only if their dot product equals 0.
- The cross product vector is perpendicular to both original vectors.
- The magnitude |u×v| equals the area of the parallelogram spanned by u and v.