Module 2: The Relationship Between Angles, Magnitudes, and the Dot Product

Consider the diagram below of two vectors:

mod23Artboard 1.png

The angle LaTeX: \thetaθ is referred to as the "angle between the vectors.'' Note that in radians LaTeX: 0 \leq \theta \leq \pi0θπ. That is the angle will always be nonnegative but not exceed LaTeX: 180^\circ180. In the extreme cases LaTeX: \theta = 0θ=0 and LaTeX: \theta = \piθ=π we say the vectors are parallel. In the case LaTeX: \theta = \pi/2θ=π/2 we say the vectors are perpendicular.

There is a very nice relation between this angle and the vectors themselves which uses the dot product. This relationship will be argued in class by means of the law of cosines:

LaTeX: |\vec{a} \cdot \vec{b}| = \|\vec{a} \| \| \vec{b} \| \cos (\theta)|ab|=abcos(θ)

where LaTeX: |\, ||| is the absolute value, LaTeX: \| \, \| is the norm, and LaTeX:  \thetaθ is the angle between vectors LaTeX: \vec{a}a and LaTeX: \vec{b}b.

This relationship is rather amazing since it says that trigonometric expressions can be defined in terms of the inner (dot) product. For example, certain collections of functions can be thought of as vectors. So we could answer the question: what is the angle between functions LaTeX: ff and LaTeX: gg?

For us, the most direct application is the following: Assuming neither LaTeX: \vec{a}a nor LaTeX: \vec{b}b is the zero vector, then:

LaTeX: \vec{a} \cdot \vec{b} = 0 \Leftrightarrow \theta = \pi/2 \Leftrightarrow \vec{a} \perp \vec{b}.ab=0θ=π/2ab.

That is: non-zero vectors are perpendicular if and only if their dot product is zero.

For example: Find all values LaTeX: tt so that the vectors

LaTeX: \begin{pmatrix}
  1\\1
  \end{pmatrix}
  \mbox{ and }
  \begin{pmatrix}
  t \\ t^2
  \end{pmatrix}(11) and (tt2)

are perpendicular.

To solve this compute their dot product and set it to zero:

LaTeX: t+t^2=0 \Leftrightarrow t(1+t) = 0.t+t2=0t(1+t)=0.

So LaTeX: t \in \left\{ -1,0\right\}t{1,0}.