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 writing 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, âÂÂaâ is the length of , and ø 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:
This article uses the convention that vectors are denoted in a bold font (e.g. ), and scalars are written in normal font (e.g. a<sub>1</sub>).
The dot product of vectors and is written as , the norm of is written âÂÂaâÂÂ, and the angle between and is denoted by ø.
The scalar projection of on is a scalar equal to
where ø is the angle between and .
A scalar projection can be used as a scale factor to compute the corresponding 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 :
By definition, the vector rejection of on is:
Hence,
When is not known, the cosine of can be computed in terms of and , by the following property of the dot product
By the above-mentioned property of the dot product, the definition of the scalar projection becomes:
In two dimensions, this becomes
Similarly, the definition of the vector projection of onto becomes:
which is equivalent to either
or
In two dimensions, the scalar rejection is equivalent to the projection of onto , which is rotated 90ð to the left. Hence,
Such a dot product is called the "perp dot product."
By definition,
Hence,
By using the Scalar rejection using the perp dot product this gives
The scalar projection on is a scalar which has a negative sign if 90 degrees < ø ⤠180 degrees. It coincides with the length of the vector projection if the angle is smaller than 90ð. More exactly:
The vector projection of on is a vector which is either null or parallel to . More exactly:
The vector rejection of on is a vector which is either null or orthogonal to . More exactly:
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:
The vector projection is an important operation in the GramâÂÂSchmidt orthonormalization of vector space bases. It is also used in the separating axis theorem to detect whether two convex shapes intersect.
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.