Vector projection
The vector projection (also known as the vector component or vector resolution) of a vector  on (or onto) a nonzero vector  is the orthogonal projection of  onto a straight line parallel to . The projection of  onto  is often written as  or . The vector component or vector resolute of  perpendicular to , sometimes also called the vector rejection of  from  (denoted  or ), is the orthogonal projection of  onto the plane (or, in general, hyperplane) that is orthogonal to . Since both  and  are vectors, and their sum is equal to , the rejection of  from  is given by: 

To simplify notation, this article defines  and 
Thus, the vector  is parallel to  the vector  is orthogonal to  and  

The projection of  onto  can be decomposed into a direction and a scalar magnitude by writting it as 
where  is a scalar, called the scalar projection of  onto , and  is the unit vector in the direction of . The scalar projection is defined as

where the operator ⋅ denotes a dot product, ‖a‖ is the length of , and θ is the angle between  and . The scalar projection is equal in absolute value to the length of the vector projection, with a minus sign if the direction of the projection is opposite to the direction of , that is, if the angle between the vectors is more than 90 degrees. The vector projection can be calculated using the dot product of  and  as:

Notation
This article uses the convention that vectors are denoted in a bold font (e.g. ), and  scalars are written in normal font (e.g. a1). The dot product of vectors  and  is written as , the norm of  is written  ‖a‖, the angle between  between  and  is denoted θ.

Definitions based on angle θ

Scalar projection

The scalar projection of  on  is a scalar equal to

where θ is the angle between  and . A scalar projection can be used as a scale factor to compute the corresponding vector projection. Vector projection
The vector projection of  on  is a vector whose magnitude is the scalar projection of  on  with the same direction as . Namely, it is defined as

where  is the corresponding scalar projection, as defined above, and  is the unit vector with the same direction as :

Vector rejection
By definition, the vector rejection of  on  is:

Hence,

Definitions in terms of a and b 
When  is not known, the cosine of  can be computed in terms of  and , by the following property of the dot product

Scalar projection
By the above-mentioned property of the dot product, the definition of the scalar projection becomes:

In two dimensions, this becomes

Vector projection
Similarly, the definition of the vector projection of  onto  becomes:

which is equivalent to either

or

Scalar rejection
In two dimensions, the scalar rejection is equivalent to the projection of  onto , which is  rotated 90° to the left. Hence,

Such a dot product is called the "perp dot product." Vector rejection
By definition,

Hence,

By using the Scalar rejection using the perp dot product this gives

Properties

Scalar projection

The scalar projection  on  is a scalar which has a negative sign if 90 degrees < θ ≤ 180 degrees. It coincides with the length  of the vector projection if the angle is smaller than 90°. More exactly:
  if ,
  if . Vector projection
The vector projection of  on  is a vector  which is either null or parallel to . More exactly:
  if ,
  and  have the same direction if ,
  and  have opposite directions if . Vector rejection
The vector rejection of  on  is a vector  which is either null or orthogonal to . More exactly:
  if  or ,
  is orthogonal to  if ,

Matrix representation
The orthogonal projection can be represented by a projection matrix. To project a vector onto the unit vector , it would need to be multiplied with this projection matrix:

Uses
The vector projection is an important operation in the Gram–Schmidt orthonormalization of vector space bases. It is also used in the separating axis theorem to detect whether two convex shapes intersect. Generalizations
Since the notions of vector length and angle between vectors can be generalized to any n-dimensional inner product space, this is also true for the notions of orthogonal projection of a vector, projection of a vector onto another, and rejection of a vector from another. In some cases, the inner product coincides with the dot product. Whenever they don't coincide, the inner product is used instead of the dot product in the formal definitions of projection and rejection. For a three-dimensional inner product space, the notions of projection of a vector onto another and rejection of a vector from another can be generalized to the notions of projection of a vector onto a plane, and rejection of a vector from a plane. The projection of a vector on a plane is its orthogonal projection on that plane. The rejection of a vector from a plane is its orthogonal projection on a straight line which is orthogonal to that plane. Both are vectors. The first is parallel to the plane, the second is orthogonal. For a given vector and plane, the sum of projection and rejection is equal to the original vector. Similarly, for inner product spaces with more than three dimensions, the notions of projection onto a vector and rejection from a vector can be generalized to the notions of projection onto a hyperplane, and rejection from a hyperplane. In geometric algebra, they can be further generalized to the notions of projection and rejection of a general multivector onto/from any invertible k-blade. See also
Scalar projection
Vector notation

References

External links
 Projection of a vector onto a plane

Operations on vectors
Transformation (function)
Functions and mappings