Fundamentals of Matrix Algebra, 2011a

5.3 Visualizing Vectors: Vectors in Three Dimensions
It is easy to compute
⎡ ⎤
2
2⃗v = ⎣ 4 ⎦ and
⎡ ⎤
−1
−⃗v = ⎣ −2 ⎦ .
6
−3
These are drawn in Figure 5.23. This figure is, in many ways, a mess, with all the
dashed and doed lines. They are useful though. Use them to see how each vector
was formed, and note that 2⃗v at least looks twice as long as ⃗v, and it looks like −⃗v
points in the opposite direcon. 20 .
Vector Length
How do we measure the length of a vector in 3D? In 2D, we were able to answer this
queson by using the Pythagorean Theorem. Does the Pythagorean Theorem apply in
3D? In a sense, it does. ⎡ ⎤
Consider the vector ⃗v = ⎣ 1 2 ⎦, as drawn in Figure 5.24 (a), with guiding dashed
3
lines. Now look at part (b) of the same figure. Note how two lengths of the dashed
lines have now been drawn gray, and another doed line has been added.
z
z
.
y
y
x
(a)
x
(b)
.
Figure 5.24: Compung the length of ⃗v
These gray dashed and doed lines form a right triangle with the doed line forming
the hypotenuse. We can find the length of the doed line using the Pythagorean
Theorem.
length of the doed line = √ sum of the squares of the dashed line lengths
That is, the length of the doed line = √ 1 2 + 2 2 = √ 5.
20 Our previous work showed that looks can be deceiving, but it is indeed true in this case.
223