Scalar projection

Summary

In mathematics, the scalar projection of a vector on (or onto) a vector also known as the scalar resolute of in the direction of is given by:

If 0° ≤ θ ≤ 90°, as in this case, the scalar projection of a on b coincides with the length of the vector projection.
Vector projection of a on b (a1), and vector rejection of a from b (a2).

where the operator denotes a dot product, is the unit vector in the direction of is the length of and is the angle between and .[1]

The term scalar component refers sometimes to scalar projection, as, in Cartesian coordinates, the components of a vector are the scalar projections in the directions of the coordinate axes.

The scalar projection is a scalar, equal to the length of the orthogonal projection of on , with a negative sign if the projection has an opposite direction with respect to .

Multiplying the scalar projection of on by converts it into the above-mentioned orthogonal projection, also called vector projection of on .

Definition based on angle θ edit

If the angle   between   and   is known, the scalar projection of   on   can be computed using

  (  in the figure)

The formula above can be inverted to obtain the angle, θ.

Definition in terms of a and b edit

When   is not known, the cosine of   can be computed in terms of   and   by the following property of the dot product  :

 

By this property, the definition of the scalar projection   becomes:

 

Properties edit

The scalar projection has a negative sign if  . It coincides with the length of the corresponding vector projection if the angle is smaller than 90°. More exactly, if the vector projection is denoted   and its length  :

  if  
  if  

See also edit

Sources edit

  • Dot products - www.mit.org
  • Scalar projection - Flexbooks.ck12.org
  • Scalar Projection & Vector Projection - medium.com
  • Lesson Explainer: Scalar Projection | Nagwa

References edit

  1. ^ Strang, Gilbert (2016). Introduction to linear algebra (5th ed.). Wellesley: Cambridge press. ISBN 978-0-9802327-7-6.