Fundamentals of Matrix Algebra, 2011a

5.3 Visualizing Vectors: Vectors in Three Dimensions
. Example 110 Find the lengths of vectors⃗v and ⃗u, where
⎡
2
⎤
⎡ ⎤
−4
⃗v = ⎣ −3 ⎦ and ⃗u = ⎣ 7 ⎦ .
5
0
S
We apply Definion 32 to each vector:
||⃗v|| =
||⃗u|| =
√
2 2 + (−3) 2 + 5 2
= √ 4 + 9 + 25
= √ 38
√
(−4) 2 + 7 2 + 0 2
= √ 16 + 49
= √ 65
.
Here we end our invesgaon into the world of graphing vectors. Extensions into
graphing 4D vectors and beyond can be done, but they truly are confusing and not
really done except for abstract purposes.
There are further things to explore, though. Just as in 2D, we can transform 3D
space by matrix mulplicaon. Doing this properly – rotang, stretching, shearing,
etc. – allows one to manipulate 3D space and create incredible computer graphics.
Exercises 5.3
In Exercises 1 – 4, vectors ⃗x and ⃗y are given.
Sketch⃗x,⃗y,⃗x +⃗y, and⃗x −⃗y on the same Cartesian
axes.
⎡ ⎤ ⎡ ⎤
1 2
1. ⃗x = ⎣ −1 ⎦,⃗y = ⎣ 3 ⎦
2 2
⎡ ⎤ ⎡ ⎤
2 −1
2. ⃗x = ⎣ 4 ⎦,⃗y = ⎣ −3 ⎦
−1 −1
⎡ ⎤ ⎡ ⎤
1 3
3. ⃗x = ⎣ 1 ⎦,⃗y = ⎣ 3 ⎦
2 6
⎡ ⎤ ⎡ ⎤
0 0
4. ⃗x = ⎣ 1 ⎦,⃗y = ⎣ −1 ⎦
1 1
In Exercises 5 – 8, vectors ⃗x and ⃗y are drawn.
Sketch 2⃗x, −⃗y, ⃗x + ⃗y, and ⃗x − ⃗y on the same
Cartesian axes.
5.
⃗y
x
z
⃗x
. y
225