Vector (What's Wrong?)

2012-07-06 11:58 pm

Let a,b be 2 vectors.|a∙b|=||a||b|cosθ| <-(θ is the angle between a,b)=|a||b||cosθ|∵ |cosθ|≥0∴ |a||b||cosθ|≥|a||b||a∙b|≥|a||b| WHAT'S WRONG?

回答 (1)

wkho28

2012-07-07 1:18 am

✔ 最佳答案

Let a, b be 2 vectors.
|a∙b|
=|a||b|cosθ|
∵ 1≥|cosθ|≥0
|a||b|≥|a||b||cosθ|≥|a||b|X0
∴ |a||b|≥|a||b||cosθ|≥0
|a∙b|≤|a||b|

2012-07-07 00:01:16 補充：
So WHAT'S WRONG with your process?

∵ |cosθ| ≥ 0 correct
∴ |a| |b| |cosθ| ≥ |a| |b| WRONG !

∵ |cosθ| ≥ 0
Both sides multiplied by |a||b|, we have
( |a| |b| ) |cosθ| ≥ ( |a| |b| ) X 0
∴ |a| |b| |cosθ| ≥ 0

2012-07-07 00:02:06 補充：
If |a| |b| |cosθ| ≥ |a| |b|
Then |cosθ| ≥ 1 which is impossible

In fact, 1 ≥ |cosθ|
Both sides multiplied by |a| |b|, we have
|a| |b| ≥ |a| |b| |cosθ|

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