Module 2: The Relationship Between Angles, Magnitudes, and the Dot Product
Consider the diagram below of two vectors:
The angle θ is referred to as the "angle between the vectors.'' Note that in radians
0≤θ≤π. That is the angle will always be nonnegative but not exceed
180∘. In the extreme cases
θ=0 and
θ=π we say the vectors are parallel. In the case
θ=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:
|→a⋅→b|=‖→a‖‖→b‖cos(θ)
where || is the absolute value,
‖‖ is the norm, and
θ is the angle between vectors
→a and
→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 f and
g?
For us, the most direct application is the following: Assuming neither →a nor
→b is the zero vector, then:
→a⋅→b=0⇔θ=π/2⇔→a⊥→b.
That is: non-zero vectors are perpendicular if and only if their dot product is zero.
For example: Find all values t so that the vectors
(11) and (tt2)
are perpendicular.
To solve this compute their dot product and set it to zero:
t+t2=0⇔t(1+t)=0.
So t∈{−1,0}.