![cover image](https://wikiwandv2-19431.kxcdn.com/_next/image?url=https://upload.wikimedia.org/wikipedia/commons/thumb/9/98/Projection_and_rejection.png/640px-Projection_and_rejection.png&w=640&q=50)
Vector projection
Concept in linear algebra / From Wikipedia, the free encyclopedia
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The vector projection (also known as the vector component or vector resolution) of a vector a 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 a perpendicular to b, sometimes also called the vector rejection of a from b (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:
![Thumb image](http://upload.wikimedia.org/wikipedia/commons/thumb/9/98/Projection_and_rejection.png/320px-Projection_and_rejection.png)
![Thumb image](http://upload.wikimedia.org/wikipedia/commons/thumb/f/f7/Projection_and_rejection_2.png/640px-Projection_and_rejection_2.png)
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: