The vector projection (also known as the vector component or vector resolution) of a vectora on (or onto) a nonzero vector b is the orthogonal projection of a onto a straight line parallel to b.
The projection of a onto b is often written as or a∥b.
The vector component or vector resolute of aperpendicular to b, sometimes also called the vector rejection of afromb (denoted or a⊥b),[1] is the orthogonal projection of a onto the plane (or, in general, hyperplane) that is orthogonal to b. Since both and are vectors, and their sum is equal to a, the rejection of a from b 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 a onto b can be decomposed into a direction and a scalar magnitude by writing it as
where is a scalar, called the scalar projection of a onto b, and b̂ is the unit vector in the direction of b. The scalar projection is defined as[2]
where the operator ⋅ denotes a dot product, ‖a‖ is the length of a, and θ is the angle between a and b.
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 b, 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. a1), and scalars are written in normal font (e.g. a1).
The dot product of vectors a and b is written as , the norm of a is written ‖a‖, the angle between a and b is denoted θ.
The scalar projection a on b is a scalar which has a negative sign if 90degrees < θ ≤ 180degrees. It coincides with the length‖c‖ of the vector projection if the angle is smaller than 90°. More exactly:
a1 = ‖a1‖ if 0° ≤ θ ≤ 90°,
a1 = −‖a1‖ if 90° < θ ≤ 180°.
Vector projection
The vector projection of a on b is a vector a1 which is either null or parallel to b. More exactly:
a1 = 0 if θ = 90°,
a1 and b have the same direction if 0° ≤ θ < 90°,
a1 and b have opposite directions if 90° < θ ≤ 180°.
Vector rejection
The vector rejection of a on b is a vector a2 which is either null or orthogonal to b. More exactly:
a2 = 0 if θ = 0° or θ = 180°,
a2 is orthogonal to b if 0 < θ < 180°,
Matrix representation
The orthogonal projection can be represented by a projection matrix.[citation needed] To project a vector onto the unit vector a = (ax, ay, az), it would need to be multiplied with this projection matrix:
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.[5] 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.
This article uses material from the Wikipedia article Vector_projection, and is written by contributors.
Text is available under a CC BY-SA 4.0 International License; additional terms may apply. Images, videos and audio are available under their respective licenses.