What we want to show: ||v|| =
2
2
2
𝑣1 + 𝑣2 + 𝑣3
Proof:
Assume v = (v1, v2, v3), with v1, v2, v3 ϵ R
Then
(v1, v2, 0) is v1 away from the origin on the x-axis
and v2 away from the origin on the y-axis
by definition of cartesian
coordinate system
So
we can construct a triangle with perpendicular legs
of lengths v1 and v2, with a hypotenuse we will call A
by definition of cartesian
coordinate
system
And
A2 = v12 + v22
A =
Then
cartesian
And
2
by Pythagorean Theorem
2
𝑣1 + 𝑣2
We can define a line B, perpendicular to A, that extends by definition of
from (v1, v2, 0) to (v1, v2, v3)
the length of B is v3, because the endpoints differ
in the Z-axis only, by v3
coordinate system
Then
And
So
We can define a line C extending from the origin to (v1, v2, v3)
C, B, and A form a right triangle
C2 = B2 + A2
by Pythagorean Theorem
2
2
2
2
C = v3 + (v1 + v2 )
by substitution
And
C=
2
2
2
𝑣1 + 𝑣2 + 𝑣3
Therefore
||v|| =
2
2
2
𝑣1 + 𝑣2 + 𝑣3
by the definition of norm