Fundamentals of Matrix Algebra, 2011a

5.3 Visualizing Vectors: Vectors in Three Dimensions
Drawing the dashed lines help us find our way in our representaon of three dimensional
space. Without them, it is hard to see how far in each direcon the vector
is supposed to have gone.
To draw ⃗u, we follow the same procedure we used to draw ⃗v. We first locate the
point (1, 3, −1), then draw the appropriate arrow. In Figure 5.15 we have ⃗u drawn
along with ⃗v. We have used different dashed and doed lines for each vector to help
disnguish them.
Noce that this me we had to go in the negave z direcon; this just means we
moved down one unit instead of up a unit.
z
. y
x
.
.
Figure 5.15: Vectors ⃗v and ⃗u in Example 103.
As in 2D, we don’t usually draw the zero vector,
⎡ ⎤
0
⃗0 = ⎣ 0 ⎦ .
0
It doesn’t point anywhere. It is a conceptually important vector that does not have a
terribly interesng visualizaon.
Our method of drawing 3D objects on a flat surface – a 2D surface – is prey clever.
It isn’t perfect, though; visually, drawing vectors with negave components (especially
negave x coordinates) can look a bit odd. Also, two very different vectors can point
to the same place. We’ll highlight this with our next two examples.
⎡
. Example 104 .Sketch the vector⃗v = ⎣ −3 ⎤
−1 ⎦.
2
S We use the same procedure we used in Example 103. Starng at
the origin, we move in the negave x direcon 3 units, then 1 unit in the negave y
direcon, and then finally up 2 units in the z direcon to find the point (−3, −1, 2).
We follow by drawing an arrow. Our sketch is found in Figure 5.16; ⃗v is drawn in two
coordinate systems, once with the helpful dashed lines, and once without. The second
drawing makes it prey clear that the dashed lines truly are helpful.
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