Vector projection

are vectors, and their sum is equal to a, the rejection of a from b is given by:

To simplify notation, this article defines

The projection of a onto b can be decomposed into a direction and a scalar magnitude by writing it as

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

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 vector projection of a on b is a vector whose magnitude is the scalar projection of a on b with the same direction as b. Namely, it is defined as

is the corresponding scalar projection, as defined above, and

is the unit vector with the same direction as b:

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

By the above-mentioned property of the dot product, the definition of the scalar projection becomes:[2]

Similarly, the definition of the vector projection of a onto b becomes:[2]

In two dimensions, the scalar rejection is equivalent to the projection of a onto

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

The scalar projection a on b is a scalar which has a negative sign if 90 degrees < θ ≤ 180 degrees.

It coincides with the length ‖c‖ of the vector projection if the angle is smaller than 90°.

To project a vector onto the unit vector a = (ax, ay, az), it would need to be multiplied with this projection matrix:

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.

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 rejection of a vector from a plane is its orthogonal projection on a straight line which is orthogonal to that plane.

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.