88
CHAPTER 1. COORDINATES AND VECTORS
Similarly,
a
1
−→
ı
= 2
vector
A
(
△O
P
1
Q
1
)
=
−→
ı
vextendsingle
vextendsingle
vextendsingle
vextendsingle
y
1
z
1
y
2
z
2
vextendsingle
vextendsingle
vextendsingle
vextendsingle
.
Finally, noting that the direction from which the
positive
z
axis is
counterclockwise
from the positive
x
axis is
−
−→
, we have
a
2
−→
= 2
vector
A
(
△O
P
2
Q
2
)
=
−
−→
vextendsingle
vextendsingle
vextendsingle
vextendsingle
x
1
z
1
x
2
z
2
vextendsingle
vextendsingle
vextendsingle
vextendsingle
.
Adding these yields the desired formula.
In each projection, we used the 2
×
2 determinant obtained by omitting the
coordinate along whose axis we were projecting. The resulting formula can
be summarized in terms of the array of coordinates of
−→
v
and
−→
w
parenleftbigg
x
1
y
1
z
1
x
2
y
2
z
2
parenrightbigg
by saying: the coefficient of the standard basis vector in a given coordinate
direction is the 2
×
2 determinant obtained by eliminating the
corresponding column from the above array, and multiplying by
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 ALL
 Calculus, Linear Algebra, Vectors, basis, Howard Staunton, standard basis vector

Click to edit the document details