What is a Vector Calculator?
This calculator computes vector addition, subtraction, dot product, cross product, and magnitude.
How it's calculated
Magnitude |v| = √(x² + y² + z²)
Dot product a·b = |a||b|cosθ (result: scalar)
Cross product a×b (result: vector)
Step-by-step example
Dot product of vectors a=(3,4) and b=(1,2)
- Step 1 Apply the formula: a·b = a₁b₁ + a₂b₂
- Step 2 Calculate: (3×1) + (4×2) = 3+8
- Step 3 Result: 11
Interpretation: Since the dot product is positive, the angle between the vectors is less than 90°.
Scalars vs. vectors
A scalar has only magnitude; a vector has both magnitude and direction. 'Traveling 60 km/h' is a scalar (speed), but 'traveling 60 km/h north' is a vector (velocity). This distinction matters in physics because direction changes the outcome entirely — two equal forces pointing the same way add up, but pointing opposite ways cancel out completely.
Good to know
- The dot product measures how aligned two vectors are — it's zero when they're perpendicular. This is exactly why physical work equals force dot displacement: a force perpendicular to motion does zero work, which is why carrying a bag horizontally means gravity technically does no work on it.
- The cross product produces a vector perpendicular to both inputs, with direction determined by the right-hand rule — torque, magnetic force, and angular momentum are all cross products.
- Vectors are the language of 3D games and graphics — character movement, lighting, and collision detection all run on vector math, and GPUs exist specifically to perform these calculations at massive scale.
- A plane flying through crosswind must angle its nose into the wind to fly a straight path — the actual flight path is the vector sum of airspeed and wind velocity, a technique called crab correction.
- The vector concept emerged from 19th-century work on Hamilton's quaternions, originally an attempt to extend complex numbers into three dimensions.
Frequently asked questions
Q. What's the difference between dot and cross product results?
Dot product gives a number (scalar); cross product gives a vector. Cross product is only defined in 3D.
Q. What does a dot product of zero mean?
The two vectors are perpendicular to each other.
Q. What is a unit vector?
A vector with magnitude 1, representing pure direction. Divide any vector by its own magnitude to get one.