🎓 Mathematics • Linear Algebra
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Vector Calculator
Performs 3D vector operations: addition, subtraction, dot product u·v, cross product u×v, magnitudes ||v||, and angles.
3D Vector Operations: Vector U & Vector V
Compute dot product, cross product, magnitudes, angles, and unit vectors.
Presets:
Vector u = ⟨ux, uy, uz⟩Magnitude ||u|| = 6.1644 • Unit = ⟨0.487, -0.324, 0.811⟩
Vector v = ⟨vx, vy, vz⟩Magnitude ||v|| = 4.5826 • Unit = ⟨0.218, 0.873, -0.436⟩
Dot Product (u · v)
-15
Angle Between Vectors (θ)122.07°2.1306 radians
Cross Product (u × v)⟨-16, 11, 14⟩Normal vector to plane
Vector Sum (u + v)⟨4, 2, 3⟩Diff: ⟨2, -6, 7⟩
What is a Vector?
A vector is a mathematical object possessing both magnitude (length) and direction.
Vectors describe forces, velocities, electric fields, and spatial coordinates across physics and engineering.
Formula & Step-by-Step Calculation
u · v = ||u|| ||v|| cos(θ), u × v = ||u|| ||v|| sin(θ) n̂
Scalar dot product and perpendicular cross product.
Worked Step-by-Step Examples
Example 1
Dot product of ⟨1, 2, 3⟩ and ⟨4, -5, 6⟩
Solution: 12
• (1×4) + (2×-5) + (3×6) = 4 - 10 + 18 = 12
Common Real-World & Academic Use Cases
- ✓ Physics work and torque calculations
- ✓ Game development 3D lighting normals
- ✓ Navigation velocity vectors
How to Use the Vector Calculator
1
Enter Vector u & v
Input coordinates ⟨x, y, z⟩.
2
Review Products
Inspect dot product, cross product, and angle θ.
Frequently Asked Questions
Q: What does a zero dot product mean?
If u · v = 0, the two vectors are perpendicular (orthogonal) to each other.