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Section 14.2 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 \uvec{u} onto a second vector \uvec{a})

the special scalar multiple of \uvec{a}\text{,}

\begin{align*} \proj_{\uvec{a}}\uvec{u} \amp= k\uvec{a}, \amp \amp\text{where} \amp k \amp= \frac{\udotprod{u}{a}}{\unorm{a}^2}; \end{align*}

sometimes called the vector component of \uvec{u} parallel to \uvec{a}

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When the initial point of \proj_{\uvec{a}}\uvec{u} is placed at the origin, the terminal point will be the point closest to \uvec{u} on the line passing through the origin and parallel to \uvec{a}\text{.}

Diagram of an orthogonal projection.
vector component of a vector \uvec{u} orthogonal to a second vector \uvec{a}

the vector \uvec{u} - \proj_{\uvec{a}}\uvec{u}

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When the initial point of the vector \uvec{u} - \proj_{\uvec{a}}\uvec{u} is placed at the terminal point of \proj_{\uvec{a}}\uvec{u}\text{,} it points towards the terminal point of \uvec{u}\text{,} at a right angle to the line that passes through the origin and is parallel to \uvec{a}\text{.} (See the diagram above.)

point-normal form (of a line in \R^2)

the vector equation \dotprod{\uvec{n}}{(\uvec{x} - \uvec{x}_0)} = 0\text{,} where \uvec{x}_0 is a vector from the origin to a known point on the line, \uvec{n} is a known normal vector for the line, and \uvec{x} is a variable vector representing an arbitrary point on the line (again as a vector from the origin)

cross product (of vectors \uvec{u} and \uvec{v} in \R^3)

a particular vector in R^3 that is orthogonal to both \uvec{u} and \uvec{v}\text{;} written \ucrossprod{u}{v}