The vector projection of a vector on (or onto) a nonzero vector , sometimes denoted
(also known as the vector component or vector resolution of in the direction of ), is the
orthogonal projection
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it wer ...
of onto a
straight line
In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are One-dimensional space, one-dimensional objects, though they may exist in Two-dimensional Euclidean space, two, Three-dimensional space, three, ...
parallel to . It is a vector parallel to , defined as:
where
is a scalar, called the
scalar projection
In mathematics, the scalar projection of a vector \mathbf on (or onto) a vector \mathbf, also known as the scalar resolute of \mathbf in the direction of \mathbf, is given by:
:s = \left\, \mathbf\right\, \cos\theta = \mathbf\cdot\mathbf,
wher ...
of onto , and is the
unit vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction vecto ...
in the direction of .
In turn, the scalar projection is defined as:
where the operator ⋅ denotes a
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
, ‖a‖ is the
length
Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...
of , and ''θ'' is the
angle
In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle.
Angles formed by two ...
between and .
Which finally gives:
The scalar projection is equal to the length of the vector projection, with a minus sign if the direction of the projection is opposite to the direction of . The vector component or vector resolute of perpendicular to , sometimes also called the vector rejection of ''from'' (denoted
), is the orthogonal projection of onto the
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* Planes (gen ...
(or, in general,
hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...
) orthogonal to . Both the projection and rejection of a vector are vectors, and their sum is equal to , which implies that the rejection is given by:
Notation
Typically, a vector projection is denoted in a bold font (e.g. ), and the corresponding scalar projection with normal font (e.g. ''a''
1). In some cases, especially in handwriting, the vector projection is also denoted using a
diacritic
A diacritic (also diacritical mark, diacritical point, diacritical sign, or accent) is a glyph added to a letter or to a basic glyph. The term derives from the Ancient Greek (, "distinguishing"), from (, "to distinguish"). The word ''diacriti ...
above or below the letter (e.g.,
or
''a''1). The vector projection of on and the corresponding rejection are sometimes denoted by and , respectively.
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
In affine geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a '' scale factor'' that is the same in all directions. The result of uniform scaling is similar ...
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
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction vecto ...
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
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
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,
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
Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...
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
In linear algebra, orthogonalization is the process of finding a set of orthogonal vectors that span a particular subspace. Formally, starting with a linearly independent set of vectors in an inner product space (most commonly the Euclidean sp ...
of
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
bases. It is also used in the
separating axis theorem
In geometry, the hyperplane separation theorem is a theorem about disjoint convex sets in ''n''-dimensional Euclidean space. There are several rather similar versions. In one version of the theorem, if both these sets are closed and at least on ...
to detect whether two convex shapes intersect.
Generalizations
Since the notions of vector
length
Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...
and
angle
In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle.
Angles formed by two ...
between vectors can be generalized to any ''n''-dimensional
inner product space
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often den ...
, 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
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often den ...
, 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
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* Planes (gen ...
, and rejection of a vector from a plane.
[ M.J. Baker, 2012]
Projection of a vector onto a plane.
Published on www.euclideanspace.com. The projection of a vector on a plane is its
orthogonal projection
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it wer ...
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
In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...
, and rejection from a
hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...
. In
geometric algebra
In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the ge ...
, they can be further generalized to the notions of
projection and rejection
The vector projection of a vector on (or onto) a nonzero vector , sometimes denoted \operatorname_\mathbf \mathbf (also known as the vector component or vector resolution of in the direction of ), is the orthogonal projection of onto a straig ...
of a general multivector onto/from any invertible ''k''-blade.
See also
*
Scalar projection
In mathematics, the scalar projection of a vector \mathbf on (or onto) a vector \mathbf, also known as the scalar resolute of \mathbf in the direction of \mathbf, is given by:
:s = \left\, \mathbf\right\, \cos\theta = \mathbf\cdot\mathbf,
wher ...
*
Vector notation
In mathematics and physics, vector notation is a commonly used notation for representing vectors, which may be Euclidean vectors, or more generally, members of a vector space.
For representing a vector, the common typographic convention is lo ...
References
External links
Projection of a vector onto a plane
{{DEFAULTSORT:Vector Projection
Operations on vectors
Transformation (function)
Functions and mappings