Question

What is the minimum value of |a + b| where and b are vectors?

Answer 1

|a + b | = ( a^2 + b^2 + 2abcos (theta) )^0.5 ..

| a + b | =( a - b )min if cos theta = -1 ....

theta = 180 ....

Answer 2

Heya user !!

Here's the answer you are looking for

We know, if a and b are 2 vectors then,

$|a + b| = \sqrt{ {a}^{2} + {b}^{2} + 2abcos \alpha }$


where \alpha is the angle between them.

The value will be min if cos\alpha will be minimum. And the minimum value of cosine function is -1 (cos 180)

So,

$a + b| = \sqrt{ {a}^{2} + {b}^{2} - 2ab }$

➡️ If a = b and the angle between the two vectors is 180°, then we get the minimum value of |a + b|, that is

$\sqrt{ {a}^{2} + {b}^{2} - 2ab} \\ = \sqrt{ {a}^{2} + {a}^{2} - 2aa } \: \: \: \: (because \: a = b) \\ = \sqrt{2 {a}^{2} - 2 {a}^{2} } \\ = \sqrt{0} = 0$


Therefore, numerically the minimum is 0.



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