4. The projection of a vector on a line

Suppose a vector v is projected onto a line.

Figure: A vector projected onto a line

The direction of the line can be expressed two ways: By giving a unit vector u along the line, or by giving an arbitrary nonzero vector w along the line, in which case we can take u$= \frac {\mbox{\bf w}} {\vert\mbox{\bf w}\vert}$.

(11)

The length of the projection of v on the line is $\vert$   v$\vert \cos \theta$ $=$ v$\cdot$   u, where a negative sign means that the projection has direction opposite to u. The length can also be expressed as $\frac {\mbox{\bf v} \cdot \mbox{\bf w}} {\vert\mbox{\bf w} \vert}$.

(12)

The vector projection of v on the line is therefore $($v$\cdot$   u$)$   u (scalar times vector). This vector projection can be called the vector component of v in the direction of the line.

(13)

If the direction of the line is expressed with an arbitrary nonzero vector w, the vector projection of v on the line becomes $\frac {(\mbox{\bf v} \cdot \mbox{\bf w}) \mbox{\bf w}} {\vert \mbox{\bf w} \vert^2}$ or $\frac {\mbox{\bf v} \cdot \mbox{\bf w}} {\mbox{\bf w} \cdot \mbox{\bf w}} \mbox{\bf w}$.

(14)

The vector component of v perpendicular to the line is just v minus the vector component along the line, so it is equal to v$- ($v$\cdot$   u$)$   u or v$- \frac {\mbox{\bf v} \cdot \mbox{\bf w}} {\mbox{\bf w} \cdot \mbox{\bf w}} \mbox{\bf w}$.

(15)

A plane is described using a normal N or unit normal n. The vector component of v perpendicular to the plane is $($v$\cdot$   n$)$   n or $\frac {\mbox{\bf v} \cdot \mbox{\bf N}} {\mbox{\bf N} \cdot \mbox{\bf N}} \mbox{\bf N}$ and the vector component of v along the plane (i.e., the projection of v on the plane) is v$- ($v$\cdot$   n$)$   n or v$- \frac {\mbox{\bf v} \cdot \mbox{\bf N}} {\mbox{\bf N} \cdot \mbox{\bf N}} \mbox{\bf N}$. These sound the other way around from () and () because N is already perpendicular to the plane.