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*} \uproj{u}{a} \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 \(\uproj{u}{a}\) 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 \(\uvec{a}\text{.}\)
A diagram consisting of several vectors in a triangular configuration to illustrate the creation of an orthogonal projection vector. The zero vector is represented by a point, and two directed line segments representing vectors labelled \(\uvec{a}\) and \(\uvec{u}\) emanate from it upwards and rightwards, with the segment for \(\uvec{u}\) longer and at a steeper ascent compared to that for \(\uvec{a}\text{,}\) so that the terminal point of \(\uvec{u}\) is higher and further to the right relative to the terminal point of \(\uvec{a}\text{.}\) In the background, a dashed line runs parallel to \(\uvec{a}\) and through the point zero vector, so that the directed line segment for vector \(\uvec{a}\) lies along the line.
A directed line segment representing a third vector runs from a point on the dashed line to the terminal point of \(\uvec{u}\) so that a right angle is formed at the initial point of this new vector, at the dashed line. Because the directed line segment for \(\uvec{u}\) is longer and steeper than that for \(\uvec{a}\text{,}\) in this scenario the initial point of the new vector necessarily falls past the terminal point of \(\uvec{a}\) on the dashed line. This new vector is labelled \(\uvec{u} - \uproj{u}{a}\text{.}\) A final directed line segment representing a vector labelled \(\uproj{u}{a}\) extends along the dashed line from the point zero vector to the initial point of \(\uvec{u} - \uproj{u}{a}\text{.}\) The three vectors \(\uvec{u}, \uproj{u}{a}, \uvec{u} - \uproj{u}{a}\) together create a right triangle.
Finally, a dotted arrow points placed inside the triangle runs from the shaft of \(\uvec{u}\) to the shaft of \(\uproj{u}{a}\text{,}\) representing the process of projecting \(\uvec{u}\) perpendicularly onto the dashed line parallel to \(\uvec{a}\) to create \(\uproj{u}{a}\text{.}\)
- vector component of a vector \(\uvec{u}\) orthogonal to a second vector \(\uvec{a}\)
- the vector \(\uvec{u} - \uproj{u}{a}\)
When the initial point of the vector \(\uvec{u} - \uproj{u}{a}\) is placed at the terminal point of \(\uproj{u}{a}\text{,}\) 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 Figure 13.2.1.)
- 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 \(\ucrossprod{u}{v}\)
