Section 13.2 Terminology and notation
- orthogonal vectors
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a pair of vectors whose dot product evaluates to \(0 \)
- normal vector (to a line or a plane)
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a vector that is orthogonal to the object of interest (the line or plane being considered)
- orthogonal projection (of vector \(\uvec{u} \) onto vector \(\uvec{a} \))
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the special scalar multiple \(\uproj{u}{a} = k \uvec{a} \) of \(\uvec{a} \) where\begin{equation*} k = \frac{\udotprod{u}{a}}{\unorm{a}^2} = \frac{\udotprod{u}{a}}{\udotprod{a}{a}} \end{equation*}
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 \(\uvec{u} \) parallel to \(\uvec{a} \)
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alternative terminology for the orthogonal projection of \(\uvec{u} \) onto \(\uvec{a} \)
- vector component of \(\uvec{u} \) orthogonal to \(\uvec{a} \)
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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 \(\R^2 \) or a plane in \(\R^3 \))
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the vector equation\begin{equation*} \dotprod{\uvec{n}}{(\uvec{x} - \uvec{x}_0)} = 0 \text{,} \end{equation*}where \(\uvec{x}_0 \) is a vector from the origin to a known point on the line/plane, \(\uvec{n} \) is a known normal vector for the line/plane, and \(\uvec{x} \) is a variable vector representing an arbitrary point on the line/plane (again as a vector from the origin)
- cross product (of vectors \(\uvec{u}, \uvec{v} \) in \(\R^3 \))
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a particular vector in \(\R^3 \) that is orthogonal to both \(\uvec{u} \) and \(\uvec{v} \text{;}\) written \(\ucrossprod{u}{v} \)
