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 \(\uvec{u}\) 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 \(\uvec{a}\)
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 \(\uvec{a}\text{.}\)
- vector component of a vector \(\uvec{u}\) orthogonal to a second vector \(\uvec{a}\)
- the vector \(\uvec{u} - \proj_{\uvec{a}}\uvec{u}\)
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)
- point-normal form (of a plane in \(\R^3\))
- 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 plane, \(\uvec{n}\) is a known normal vector for the plane, and \(\uvec{x}\) is a variable vector representing an arbitrary point on the plane (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 \(\ucrossprod{u}{v}\)
