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# Perpendicular Vector

A vector perpendicular to a given vector is a vector (voiced "-perp") such that and form a right angle.

In the plane, there are two vectors perpendicular to any given vector, one rotated counterclockwise and the other rotated clockwise. Hill (1994) defines to be the perpendicular vector obtained from an initial vector

 (1)

by a counterclockwise rotation by , i.e.,

 (2)

In the plane, a vector perpendicular to can therefore be obtained by transposing the Cartesian components and taking the minus sign of one. This operation is implemented in the Wolfram Language as Cross[ax, ay].

In three dimensions, there are an infinite number of vectors perpendicular to a given vector, all satisfying the equations

 (3)

Perp Dot Product, Perpendicular, Vector

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## References

Hill, F. S. Jr. "The Pleasures of 'Perp Dot' Products." Ch. II.5 in Graphics Gems IV (Ed. P. S. Heckbert). San Diego: Academic Press, pp. 138-148, 1994.

## Referenced on Wolfram|Alpha

Perpendicular Vector

## Cite this as:

Weisstein, Eric W. "Perpendicular Vector." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PerpendicularVector.html