DLA Terminology and notation

Section 13.2 Terminology and notation

orthogonal vectors: a pair of vectors whose dot product evaluates to $0$
🔗
normal vector (to a line or a plane): a vector that is orthogonal to the object of interest (i.e. the line or plane being considered)
🔗
orthogonal projection (of a vector $u$ onto a second vector $a$ ): the special scalar multiple of $a,$
🔗

$\begin{aligned} {proj}_{a} u & = k a, & where & k & = \frac{u \cdot a}{{‖ a ‖}^{2}}; \end{aligned}$

sometimes called the vector component of $u$ parallel to $a$
🔗

🔗

When the initial point of

{proj}_{a} u

is placed at the origin, the terminal point will be the point closest to

u

on the line passing through the origin and parallel to

a .

Diagram of an orthogonal projection.

vector component of a vector $u$ orthogonal to a second vector $a$: the vector $u - {proj}_{a} u$
🔗

When the initial point of the vector

u - {proj}_{a} u

is placed at the terminal point of

{proj}_{a} u,

it points towards the terminal point of

u,

at a right angle to the line that passes through the origin and is parallel to

a .

(See the diagram above.)

point-normal form (of a line in $R^{2}$ ): the vector equation $n \cdot (x - x_{0}) = 0,$ where $x_{0}$ is a vector from the origin to a known point on the line, $n$ is a known normal vector for the line, and $x$ is a variable vector representing an arbitrary point on the line (again as a vector from the origin)
🔗
point-normal form (of a plane in $R^{3}$ ): the vector equation $n \cdot (x - x_{0}) = 0,$ where $x_{0}$ is a vector from the origin to a known point on the plane, $n$ is a known normal vector for the plane, and $x$ is a variable vector representing an arbitrary point on the plane (again as a vector from the origin)
🔗
cross product (of vectors $u$ and $v$ in $R^{3}$ ): a particular vector in $R^{3}$ that is orthogonal to both $u$ and $v;$ written $u \times v$
🔗