Vector projection

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 straight line parallel to . It is a vector parallel to , defined as:
$\mathbf_1 = a_1\mathbf$
where $a_1$ is a scalar, called the scalar projection of onto , and is the unit vector in the direction of .

In turn, the scalar projection is defined as:
$a_1 = \left\, \mathbf\right\, \cos\theta = \mathbf\cdot\mathbf$
where the operator ⋅ denotes a dot product, ‖a‖ is the length of , and ''θ'' is the angle between and .

Which finally gives:
$\mathbf_1 = \left(\mathbf \cdot \mathbf\right) \mathbf = \frac \frac = \frac = \frac ~ .$

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 $\operatorname_ \mathbf$), is the orthogonal projection of onto the plane (or, in general, hyperplane) 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: $\mathbf_2 = \mathbf - \mathbf_1.$

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 above or below the letter (e.g., $\vec_1$ 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
$a_1 = \left\, \mathbf\right\, \cos \theta ,$
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
$\mathbf_1 = a_1 \mathbf = (\left\, \mathbf\right\, \cos \theta) \mathbf$
where $a_1$ is the corresponding scalar projection, as defined above, and $\mathbf$ is the unit vector with the same direction as :
$\mathbf = \frac$

Vector rejection

By definition, the vector rejection of on is:
$\mathbf_2 = \mathbf - \mathbf_1$

Hence,
$\mathbf_2 = \mathbf - \left(\left\, \mathbf\right\, \cos \theta\right) \mathbf$

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
$\frac = \cos \theta$

Scalar projection

By the above-mentioned property of the dot product, the definition of the scalar projection becomes:
$a_1 = \left\, \mathbf\right\, \cos \theta = \left\, \mathbf\right\, \frac = \frac .$

In two dimensions, this becomes
$a_1 = \frac .$

Vector projection

Similarly, the definition of the vector projection of onto becomes:
$\mathbf_1 = a_1 \mathbf = \frac \frac ,$
which is equivalent to either
$\mathbf_1 = \left(\mathbf \cdot \mathbf\right) \mathbf,$
or
$\mathbf_1 = \frac = \frac ~ .$

Scalar rejection

In two dimensions, the scalar rejection is equivalent to the projection of onto $\mathbf^\perp = \begin-\mathbf_y & \mathbf_x\end$, which is $\mathbf = \begin\mathbf_x & \mathbf_y\end$ rotated 90° to the left. Hence,
$a_2 = \left\, \mathbf\right\, \sin \theta = \frac = \frac .$

Such a dot product is called the "perp dot product."

Vector rejection

By definition,
$\mathbf_2 = \mathbf - \mathbf_1$

Hence,
$\mathbf_2 = \mathbf - \frac .$

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:
$P_\mathbf = \mathbf \mathbf^\textsf = \begin a_x \\ a_y \\ a_z \end \begin a_x & a_y & a_z \end = \begin a_x^2 & a_x a_y & a_x a_z \\ a_x a_y & a_y^2 & a_y a_z \\ a_x a_z & a_y a_z & a_z^2 \\ \end$

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. 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 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

{{DEFAULTSORT:Vector Projection

Operations on vectors
Transformation (function)
Functions and mappings