Perpendicular Vector

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


In the plane, there are two vectors perpendicular to any given vector, one rotated 90 degrees counterclockwise and the other rotated 90 degrees clockwise. Hill (1994) defines a^_|_ to be the perpendicular vector obtained from an initial vector

 a=[a_x; a_y]

by a counterclockwise rotation by 90 degrees, i.e.,

 a^_|_=[0 -1; 1 0]a=[-a_y; a_x].

In the plane, a vector perpendicular to a=(a_x,a_y) 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


See also

Perp Dot Product, Perpendicular, Vector

Explore with Wolfram|Alpha


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.

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