vector projection

The principle used in the projection of line segment a line, which results a line segmentMathworldPlanetmath, may be extended to concern the projection of a vector u on another non-zero vector v, resulting a vector.

This projection vector, the so-called vector projectionuv  will be to v.  It could have the length ( equal to |u| multiplied by the cosine of the inclination angle between the lines of u and v, as in the case of line segment.

But better than that “inclination angle” is to take the between the both vectors u and v which may also be obtuse or straight; in these cases the cosine is negative which is suitable to cause the projection vector uv to have the direction to v  (uvv).  In all cases we define the vector projection or the vector component of u along v as

uv:=|u|cos(u,v)v (1)

where v is the unit vectorMathworldPlanetmath having the direction as v  (i.e., vv).  For the that if  u=0  and the angle is , then also the vector projection is the zero vectorMathworldPlanetmath.

Using the expression for the of the angle between vectors and for the unit vector we thus have


This is to

uv=uv|v||v|v, (2)

where the denominator is the scalar square of v:

uv=uvvvv (3)

One can also write from (1) the alternative form

uv=(uv)v, (4)

where the “coefficient” uv of the unit vector v is called the scalar projection or the scalar component of u along v.

Remark 1.  The vector projection  uv  of u along v is sometimes denoted by  projvu.

Remark 2.  If one subtracts ( from u the vector component uv, then one has another componentPlanetmathPlanetmathPlanetmath of u such that the both components are orthogonalMathworldPlanetmathPlanetmath to each other (and their sum ( is u); the orthogonality of the components follows from

(u-uv)uv=uvvvuv-(uvvv)2vv= 0.

Remark 3.  The usual “component form”


of vectors in the cartesian coordinate system of 3 that the orthogonal ( vector components of u along the unit vectors i, j, k are


and the scalar components are x, y, z, respectively.

Title vector projection
Canonical name VectorProjection
Date of creation 2013-03-22 19:05:40
Last modified on 2013-03-22 19:05:40
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 13
Author pahio (2872)
Entry type Definition
Classification msc 51N99
Classification msc 51M04
Classification msc 51F20
Related topic Projection
Related topic GramSchmidtOrthogonalization
Defines vector component
Defines scalar projection
Defines scalar component