Fundamentals of Matrix Algebra, 2011a

2.3 Visualizing Matrix Arithmec in 2D
.
⃗y
⃗x
⃗y
⃗x
⃗x −⃗y
(a)
(b)
.
.
Figure 2.13: Vectors ⃗x, ⃗y and ⃗x − ⃗y in Example 37
Vector Length
When we discussed scalar mulplicaon, we made reference to a fundamental
queson: How do we measure the length of a vector? Basic geometry gives us an
answer in the two dimensional case that we are dealing with right now, and later we
can extend these ideas to higher dimensions.
Consider Figure 2.14. A vector ⃗x is drawn in black, and dashed and doed lines
have been drawn to make it the hypotenuse of a right triangle.
⃗x
1
.
1
.
Figure 2.14: Measuring the length of a vector
It is easy to see that the dashed line has length 4 and the doed line has length 3.
We’ll let c denote the length of⃗x; according to the Pythagorean Theorem, 4 2 +3 2 = c 2 .
Thus c 2 = 25 and we quickly deduce that c = 5.
Noce that in our figure, ⃗x goes to the right 4 units and then up 3 units. In other
words, we can write
[ ] 4
⃗x = .
3
We learned above that the length of ⃗x is √ 4 2 + 3 2 . 10 This hints at a basic calculaon
that works for all vectors⃗x, and we define the length of a vector according to this rule.
10 Remember that √ 4 2 + 3 2 ≠ 4 + 3!
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