A First Course in Linear Algebra, 2017a

156 R n = √ 1 + 9 + 16
= √ 26
In order to find ⃗u, wedivide⃗v by √ 26. The result is
⃗u = 1
‖⃗v‖ ⃗v
=
=
1
√
26
[
1 −3 4
] T
[
]
√1
− √ 3 4 T
√
26 26 26
You can verify using the Definition 4.14 that ‖⃗u‖ = 1.
♠
4.5 Geometric Meaning of Scalar Multiplication
Outcomes
A. Understand scalar multiplication, geometrically.
Recall that
√
the point P =(p 1 , p 2 , p 3 ) determines a vector ⃗p from 0 to P. The length of ⃗p, denoted ‖⃗p‖,
is equal to p 2 1 + p2 2 + p2 3
by Definition 4.10.
Now suppose we have a vector ⃗u = [ ] T
u 1 u 2 u 3 and we multiply ⃗u by a scalar k. By Definition
4.5, k⃗u = [ ] T
ku 1 ku 2 ku 3 . Then, by using Definition 4.10, the length of this vector is given by
√ (
(ku 1 ) 2 +(ku 2 ) 2 +(ku 3 ) 2) = |k|√
u 2 1 + u2 2 + u2 3
Thus the following holds.
‖k⃗u‖ = |k|‖⃗u‖
In other words, multiplication by a scalar magnifies or shrinks the length of the vector by a factor of |k|.
If |k| > 1, the length of the resulting vector will be magnified. If |k| < 1, the length of the resulting vector
will shrink. Remember that by the definition of the absolute value, |k|≥0.
What about the direction? Draw a picture of ⃗u and k⃗u where k is negative. Notice that this causes the
resulting vector to point in the opposite direction while if k > 0 it preserves the direction the vector points.
Therefore the direction can either reverse, if k < 0, or remain preserved, if k > 0.
Consider the following example.