vector projection

The principle used in the projection of line segment a line, which results a line segment, may be extended to concern the projection of a vector $\overrightarrow{u}$ on another non-zero vector $\overrightarrow{v}$, resulting a vector.

This projection vector, the so-called vector projection  ${\overrightarrow{u}}_{\overrightarrow{v}}$  will be http://planetmath.org/node/6178parallel to $\overrightarrow{v}$.  It could have the length (http://planetmath.org/Vector) equal to $|\overrightarrow{u}|$ multiplied by the cosine of the inclination angle between the lines of $\overrightarrow{u}$ and $\overrightarrow{v}$, 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 $\overrightarrow{u}$ and $\overrightarrow{v}$ which may also be obtuse or straight; in these cases the cosine is negative which is suitable to cause the projection vector ${\overrightarrow{u}}_{\overrightarrow{v}}$ to have the direction to $\overrightarrow{v}$  (${\overrightarrow{u}}_{\overrightarrow{v}}\downarrow \uparrow \overrightarrow{v}$).  In all cases we define the vector projection or the vector component of $\overrightarrow{u}$ along $\overrightarrow{v}$ as

${\overrightarrow{u}}_{\overrightarrow{v}}:=|\overrightarrow{u}|\cos (\overrightarrow{u},\overrightarrow{v}){\overrightarrow{v}}^{\circ }$ (1)

where ${\overrightarrow{v}}^{\circ }$ is the unit vector having the http://planetmath.org/node/6178same direction as $\overrightarrow{v}$  (i.e., ${\overrightarrow{v}}^{\circ }\upuparrows \overrightarrow{v}$).  For the that if  $\overrightarrow{u}=\overrightarrow{0}$  and the angle is , then also the vector projection is the zero vector.

Using the expression for the http://planetmath.org/node/6178cosine of the angle between vectors and for the unit vector we thus have

$${\overrightarrow{u}}_{\overrightarrow{v}}=|\overrightarrow{u}|\frac{\overrightarrow{u}\cdot \overrightarrow{v}}{|\overrightarrow{u}||\overrightarrow{v}|}\frac{\overrightarrow{v}}{|\overrightarrow{v}|}.$$

This is to

${\overrightarrow{u}}_{\overrightarrow{v}}={\displaystyle \frac{\overrightarrow{u}\cdot \overrightarrow{v}}{|\overrightarrow{v}||\overrightarrow{v}|}}\overrightarrow{v},$ (2)

where the denominator is the scalar square of $\overrightarrow{v}$:

${\overrightarrow{u}}_{\overrightarrow{v}}={\displaystyle \frac{\overrightarrow{u}\cdot \overrightarrow{v}}{\overrightarrow{v}\cdot \overrightarrow{v}}}\overrightarrow{v}$ (3)

One can also write from (1) the alternative form

${\overrightarrow{u}}_{\overrightarrow{v}}=(\overrightarrow{u}\cdot {\overrightarrow{v}}^{\circ }){\overrightarrow{v}}^{\circ },$ (4)

where the “coefficient” $\overrightarrow{u}\cdot {\overrightarrow{v}}^{\circ }$ of the unit vector ${\overrightarrow{v}}^{\circ }$ is called the scalar projection or the scalar component of $\overrightarrow{u}$ along $\overrightarrow{v}$.

Remark 1.  The vector projection  ${\overrightarrow{u}}_{\overrightarrow{v}}$  of $\overrightarrow{u}$ along $\overrightarrow{v}$ is sometimes denoted by  ${\text{proj}}_{\overrightarrow{v}}\overrightarrow{u}$.

Remark 2.  If one subtracts (http://planetmath.org/DifferenceOfVectors) from $\overrightarrow{u}$ the vector component ${\overrightarrow{u}}_{\overrightarrow{v}}$, then one has another component of $\overrightarrow{u}$ such that the both components are orthogonal to each other (and their sum (http://planetmath.org/SumVector) is $\overrightarrow{u}$); the orthogonality of the components follows from

$$(\overrightarrow{u}-{\overrightarrow{u}}_{\overrightarrow{v}})\cdot {\overrightarrow{u}}_{\overrightarrow{v}}=\frac{\overrightarrow{u}\cdot \overrightarrow{v}}{\overrightarrow{v}\cdot \overrightarrow{v}}\overrightarrow{u}\cdot \overrightarrow{v}-{\left(\frac{\overrightarrow{u}\cdot \overrightarrow{v}}{\overrightarrow{v}\cdot \overrightarrow{v}}\right)}^2\overrightarrow{v}\cdot \overrightarrow{v}=\mathrm{\hspace{0.33em}0}.$$

Remark 3.  The usual “component form”

$$\overrightarrow{u}=x\overrightarrow{i}+y\overrightarrow{j}+z\overrightarrow{k}$$

of vectors in the cartesian coordinate system of ${ℝ}^3$ that the orthogonal (http://planetmath.org/OrthogonalVectors) vector components of $\overrightarrow{u}$ along the unit vectors $\overrightarrow{i}$, $\overrightarrow{j}$, $\overrightarrow{k}$ are

$${\overrightarrow{u}}_{\overrightarrow{i}}=x\overrightarrow{i},{\overrightarrow{u}}_{\overrightarrow{j}}=y\overrightarrow{j},{\overrightarrow{u}}_{\overrightarrow{k}}=z\overrightarrow{k}$$

and the scalar components are $x$, $y$, $z$, respectively.

Title	vector projection
Canonical name	VectorProjection
Date of creation	2013-03-22 19:05:40
Last modified on	2013-03-22 19:05:40
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	13
Author	pahio (2872)
Entry type	Definition
Classification	msc 51N99
Classification	msc 51M04
Classification	msc 51F20
Related topic	Projection
Related topic	GramSchmidtOrthogonalization
Defines	vector component
Defines	scalar projection
Defines	scalar component