Page 106 - A Course in Linear Algebra with Applications

90 Chapter Four: Introduction to Vector Spaces

and their sum

A + B = a 2 + b 2 .
\ a 3 + b J
3
Represent the vectors A, B and A + B by line segments IU,
IV, and W in three dimensional space with a common initial
I
point I (ui, u 2, u 3), say. The line segments determine a figure
I U W V as shown:

where U, W and V are the points

(ui+ai, u 2+a 2, u 3+a 3), (ui+ai+bi, u 2+a 2+b 2, u 3+a 3+b 3)

and
(ui +h, u 2 +b 2, u 3 + b 3),

respectively.
In fact U W V is a parallelogram. To prove this, we need
I
to find the lengths and directions of the four sides. Simple
analytic geometry shows that IU'= VW = \/a\ + | + a§ = /,
a
and that IV = UW = ^Jb\ + b\ + bj = m, say. Also the
direction cosines of U and V W are ai/l, a 2/l, a 3/l, while
I
those of IV and U W are bi/m, b 2/m, b 3/m. It follows that
I
opposite sides of U W V are parallel and of equal length, so
it is indeed a parallelogram.
These considerations show that the rule of addition for
vectors in R 3 is equivalent to the parallelogram rule for addi-
tion of forces, which is familiar from mechanics. To add line