The vector projection (also known as the vector componentorvector resolution) of a vectora on (or onto) a nonzero vector b is the orthogonal projectionofa onto a straight line parallel to b.
The projection of a onto b is often written as ora∥b.
The vector component or vector resolute of aperpendiculartob, sometimes also called the vector rejectionofafromb (denoted ora⊥b),[1] is the orthogonal projection of a onto the plane (or, in general, hyperplane) that is orthogonaltob. 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 projectionofa 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 lengthofa, 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:
The vector projection of aonb is a vector whose magnitude is the scalar projection of aonb with the same direction as b. Namely, it is defined as
where is the corresponding scalar projection, as defined above, and is the unit vector with the same direction as b:
The scalar projection aonb 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°. More exactly:
The orthogonal projection can be represented by a projection matrix. 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.