Establish the following vector inequalities geometrically or
otherwise:
(a) $| a + b | \leq | a | + | b |$
(b) $| a + b | \geq | | a | - | b | |$

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Solution

$F r o m t r i a n g l e l a w o f v e c t o r s :$
$¯ ¯ c = ¯ ¯ ¯ a + ¯ ¯ b$
$F r o m t r i a n g l e p r o p e r t y :$
$w e k n o w s u m o f t w o s i d e s \geq t h i r d s i d e .$
$⟹ ∣ ∣ ¯ ¯ ¯ a ∣ ∣ + ∣ ∣ ¯ ¯ b ∣ ∣ \geq ∣ ∣ ¯ ¯ c ∣ ∣$
$⟹ ∣ ∣ ¯ ¯ ¯ a ∣ ∣ + ∣ ∣ ¯ ¯ b ∣ ∣ \geq ∣ ∣ ¯ ¯ ¯ a + ¯ ¯ b ∣ ∣$
$⟹ ∣ ∣ ¯ ¯ ¯ a + ¯ ¯ b ∣ ∣ \leq ∣ ∣ ¯ ¯ ¯ a ∣ ∣ + ∣ ∣ ¯ ¯ b ∣ ∣$
$∴ a) i s e s t a b l i s h e d .$
$S i m i l a r i l y, w e k n o w f r o m t r i a n g l e p r o p e r t y :$
$D i f f e r e n c e o f t w o s i d e s \leq t h i r d s i d e .$
$⟹ ∣ ∣ ∣ ∣ ¯ ¯ ¯ a ∣ ∣ - ∣ ∣ ¯ ¯ b ∣ ∣ ∣ ∣ \leq ∣ ∣ ¯ ¯ c ∣ ∣$
$⟹ ∣ ∣ ∣ ∣ ¯ ¯ ¯ a ∣ ∣ - ∣ ∣ ¯ ¯ b ∣ ∣ ∣ ∣ \leq ∣ ∣ ¯ ¯ ¯ a + ¯ ¯ b ∣ ∣$
$⟹ ∣ ∣ ¯ ¯ ¯ a + ¯ ¯ b ∣ ∣ \geq ∣ ∣ ∣ ∣ ¯ ¯ ¯ a ∣ ∣ - ∣ ∣ ¯ ¯ b ∣ ∣ ∣ ∣$
$∴ b) i s e s t a b l i s h e d .$

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Similar questions

Establish the following vector inequalities geometrically or otherwise:

(a) |a + b| ≤ |a| + |b|

(b) |a + b| ≥ ||a| − |b||

(d) |a − b| ≥ ||a| − |b||

When does the equality sign above apply?

\to a

\to b

| \to a + \to b | \leq | \to a | + | \to b |

| \to a - \to b | \geq | \to a | - | \to b |

\to a

\to b

∣ ∣ ∣ \to a + \to b ∣ ∣ ∣ \leq ∣ ∣ \to a ∣ ∣ + ∣ ∣ ∣ \to b ∣ ∣ ∣

: | ¯ ¯ ¯ a + ¯ ¯ b | = | ¯ ¯ ¯ a - ¯ ¯ b | \Leftrightarrow ¯ ¯ ¯ a ⊥ ¯ ¯ b

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