Math Tools

Vector Addition

Vector addition is very straightforward. For two vectors whose dimensions match, write them left-to-right, one over the other. For each component, the corresponding component in the sum vector is just the sum of those same components in the original vector.

To use the calculator, add vectors to the calculator until you reach the desired number. Then set the number of dimensions that each of the vectors should have.

The resultant vector is listed at the bottom.

# of Vectors

Dimensions

Result

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Scalar Multiplication

Multiplying a vector by a scalar is very straightforward. For any given component in the vector, the corresponding component in the output vector is that component multiplied by the scalar.

Scalar:

Dimensions

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Result

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Vector Magnitude

Getting the magnitude of a vector involves extending the Pythagorean theorem into an arbitrary number of dimensions. Take all of the components, square each of them, add them together, then take the square root.

Dimensions

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Magnitude:

Unit Vector

Finding the unit vector along a given vector's axis involves first getting that vector's magnitude. Then divide each component by that magnitude.

Dimensions

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Result

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Dot Product

To take the dot product of two vectors whose dimensions match, sum together the products of each corresponding component.

Dimensions

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Dot Product:

Cross Product

Defined for 3D vectors: a × b is perpendicular to both a and b.

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Result

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Triple Scalar Product

a · (b × c). Signed volume of the parallelepiped spanned by a, b, c (3D).

a · (b × c):

Vector Projection

proj_b(a) = (a · b / ||b||²) b

Dimensions

a = < >

b = < >

Scalar component (a · b / ||b||):

Projection vector

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