A First Course in Linear Algebra, 2017a

4.7. The Dot Product 169
Example 4.34: Using Geometric Description to Find a Dot Product
Let ⃗u,⃗v be vectors with ‖⃗u‖ = 3 and ‖⃗v‖ = 4. Suppose the angle between ⃗u and⃗v is π/3. Find⃗u•⃗v.
Solution. From the geometric description of the dot product in Proposition 4.31
⃗u •⃗v =(3)(4)cos(π/3)=3 × 4 × 1/2 = 6
Two nonzero vectors are said to be perpendicular, sometimes also called orthogonal, if the included
angle is π/2radians(90 ◦ ).
Consider the following proposition.
Proposition 4.35: Perpendicular Vectors
Let ⃗u and⃗v be nonzero vectors in R n . Then, ⃗u and⃗v are said to be perpendicular exactly when
⃗u •⃗v = 0
♠
Proof. This follows directly from Proposition 4.31. First if the dot product of two nonzero vectors is equal
to 0, this tells us that cosθ = 0 (this is where we need nonzero vectors). Thus θ = π/2 and the vectors are
perpendicular.
If on the other hand ⃗v is perpendicular to ⃗u, then the included angle is π/2 radians. Hence cosθ = 0
and ⃗u •⃗v = 0.
♠
Consider the following example.
Example 4.36: Determine if Two Vectors are Perpendicular
Determine whether the two vectors,
⎡ ⎤ ⎡ ⎤
2 1
⃗u = ⎣ 1 ⎦,⃗v = ⎣ 3 ⎦
−1 5
are perpendicular.
Solution. In order to determine if these two vectors are perpendicular, we compute the dot product. This
is given by
⃗u •⃗v =(2)(1)+(1)(3)+(−1)(5)=0
Therefore, by Proposition 4.35 these two vectors are perpendicular.
♠