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Math Help - vector

  1. #1
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    vector

    For any 2 vectors a and b , |a| + |b| is always greater than or equal to |a+b| .

    Can someone pls explain the statement above
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  2. #2
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    |a| means the modulus of the vector a, or basically if it is postive it stays the same, and if it is negative u multiply it by -1.

    If a,b \leq 0 then:
    |a|+|b| = |a+b|

    if a,b \geq 0 then:
    |a|+|b| = |a+b|

    However if a < 0 and b > 0 or b < 0 and a > 0 then:
    |a|+|b| > |a+b|

    So thats why For any 2 vectors a and b , |a| + |b| is always greater than or equal to |a+b| .
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  3. #3
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    Quote Originally Posted by mathaddict View Post
    For any 2 vectors a and b , |a| + |b| is always greater than or equal to |a+b| .

    Can someone pls explain the statement above
    You can prove this by Cauchy-Schwartz inequality: |\bold{a}\cdot \bold{b}| \leq |\bold{a}|\cdot |\bold{b}|.

    |\bold{a}+\bold{b}|^2 = (\bold{a}+\bold{b})\cdot (\bold{a}+\bold{b}) = |\bold{a}|^2 + |\bold{b}|^2 + 2\bold{a}\cdot \bold{b} \leq (|\bold{a}|+|\bold{b}|)^2

    Now take square roots to get,
    |\bold{a}+\bold{b}| \leq |\bold{a}|+|\bold{b}|
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