A First Course in Linear Algebra, 2017a

164 R n
Example 4.26: Compute a Dot Product
Find ⃗u •⃗v for
⃗u =
⎡
⎢
⎣
1
2
0
−1
⎤
⎡
⎥
⎦ ,⃗v = ⎢
⎣
0
1
2
3
⎤
⎥
⎦
Solution. By Definition 4.25, we must compute
This is given by
⃗u •⃗v =
4
∑ u k v k
k=1
⃗u •⃗v = (1)(0)+(2)(1)+(0)(2)+(−1)(3)
= 0 + 2 + 0 + −3
= −1
With this definition, there are several important properties satisfied by the dot product.
♠
Proposition 4.27: Properties of the Dot Product
Let k and p denote scalars and ⃗u,⃗v,⃗w denote vectors. Then the dot product ⃗u •⃗v satisfies the following
properties.
• ⃗u •⃗v =⃗v •⃗u
• ⃗u •⃗u ≥ 0 and equals zero if and only if ⃗u =⃗0
• (k⃗u + p⃗v) •⃗w = k (⃗u •⃗w)+p(⃗v •⃗w)
• ⃗u • (k⃗v + p⃗w)=k (⃗u •⃗v)+p(⃗u •⃗w)
• ‖⃗u‖ 2 =⃗u •⃗u
The proof is left as an exercise. This proposition tells us that we can also use the dot product to find
the length of a vector.
Example 4.28: Length of a Vector
Find the length of
That is, find ‖⃗u‖.
⃗u =
⎡
⎢
⎣
2
1
4
2
⎤
⎥
⎦