This projection vector, the so-called vector projection will be http://planetmath.org/node/6178parallel to . It could have the length (http://planetmath.org/Vector) equal to multiplied by the cosine of the inclination angle between the lines of and , as in the case of line segment.
But better than that “inclination angle” is to take the http://planetmath.org/node/6178angle between the both vectors and which may also be obtuse or straight; in these cases the cosine is negative which is suitable to cause the projection vector to have the direction to (). In all cases we define the vector projection or the vector component of along as
Using the expression for the http://planetmath.org/node/6178cosine of the angle between vectors and for the unit vector we thus have
This is to
where the denominator is the scalar square of :
One can also write from (1) the alternative form
where the “coefficient” of the unit vector is called the scalar projection or the scalar component of along .
Remark 1. The vector projection of along is sometimes denoted by .
Remark 2. If one subtracts (http://planetmath.org/DifferenceOfVectors) from the vector component , then one has another component of such that the both components are orthogonal to each other (and their sum (http://planetmath.org/SumVector) is ); the orthogonality of the components follows from
Remark 3. The usual “component form”
of vectors in the cartesian coordinate system of that the orthogonal (http://planetmath.org/OrthogonalVectors) vector components of along the unit vectors , , are
and the scalar components are , , , respectively.
|Date of creation||2013-03-22 19:05:40|
|Last modified on||2013-03-22 19:05:40|
|Last modified by||pahio (2872)|