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.