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4.3 Directions and Magnitudes 93
Theorem 4.3.2 (Triangle Inequality). For any u, v ∈ R n
Proof.
‖u + v‖ ≤ ‖u‖ + ‖v‖.
‖u + v‖ 2 = (u + v) (u + v)
= u u + 2u v + v v
= ‖u‖ 2 + ‖v‖ 2 + 2 ‖u‖ ‖v‖ cos θ
= (‖u‖ + ‖v‖) 2 + 2 ‖u‖ ‖v‖(cos θ − 1)
≤ (‖u‖ + ‖v‖) 2 .
That is, the square of the left-hand side of the triangle inequality is ≤ the
square of the right-hand side. Since both the things being squared are positive,
the inequality holds without the square;
‖u + v‖ ≤ ‖u‖ + ‖v‖
The triangle inequality is also “self-evident” when examining a sketch of
u, v and u + v.
Example 54 Let
so that
⎛ ⎞ ⎛ ⎞
1
4
a = ⎜2
⎟
⎝3⎠ and b = ⎜3
⎟
⎝2⎠ ,
4
1
a a = b b = 1 + 2 2 + 3 2 + 4 2 = 30
93