Vector Analysis Calculators
This category presents calculators describing vector behaviour in multiple dimensions. Each tool provides numerical outputs for magnitudes, directional angles, component relationships and product values across a range of vector operations.
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Angle Between Two Vectors Calculator
The angle between two vectors depends on how closely their directions align, creating a clear measure of how similar or opposite their paths are.
Example use: Comparing the direction of two thrown balls to see how far apart their paths spread.
Inputs: vector u components, vector v components
Outputs: magnitude of u, magnitude of v, dot product, scalar projection, determinant as two-dimensional cross product, parallelogram area, resultant vector as u plus v, vector relationship, angle in radians, angle in degrees
Visual: two vectors drawn from the same starting point with the resultant vector and the angle between them highlighted
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Dot Product and Cross Product Calculator
The dot and cross products describe how two vectors relate, showing how much they point in the same direction and how much area they span together.
Example use: Checking how similar two walking directions are when comparing steps taken by two people.
Inputs: vector u components, vector v components
Outputs: dot product, cross product as vertical component, parallelogram area, vector addition, distance between heads, angle, unit vectors, projection of u onto v, interpretation
Visual: two vectors shown from a shared starting point with their relative direction emphasised
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Magnitude and Unit Vector Calculator
The magnitude of a vector gives its overall length, while the unit vector shows the same direction scaled down to a length of one.
Example use: Comparing the direction of a breeze without focusing on how strong it is.
Inputs: component vx, component vy, component vz
Outputs: magnitude of v, unit vector, direction angle to x-axis, direction angle to y-axis, direction angle to z-axis, azimuth angle
Visual: a three-dimensional vector with a shorter unit vector pointing the same way
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Vector Addition and Subtraction Calculator
Adding or subtracting vectors combines their horizontal and vertical changes to produce a new direction and length.
Example use: Working out the net direction of two separate pushes applied to a toy on the floor.
Inputs: vector u components, vector v components, operation choice
Outputs: resultant components, resultant magnitude, resultant direction angle, unit vector, quadrant, input vector magnitudes and angles
Visual: two input vectors and the resulting vector shown together from a shared starting point
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Vector Projection Calculator
The projection of one vector onto another shows how much of its direction lies along the second vector, with the remainder forming the rejection.
Example use: Checking how much of a person's walking direction lines up with a straight footpath.
Inputs: vector a components, vector b components
Outputs: scalar projection, vector projection, vector rejection, angle between vectors, dot product, magnitude of b
Visual: two vectors with the projection drawn along one vector and the rejection shown at a right angle
Vector Analysis FAQ
A scalar describes size only, while a vector describes both size and direction, representing orientation or movement in space.
The dot product presents alignment and projection values, while the cross product generates a perpendicular vector describing rotational or directional behaviour.
A unit vector has length one and describes direction only, providing orientation without affecting the magnitude of related quantities.