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If |bar(a)+bar(b)| = |bar(a)-bar(b)| the...

If `|bar(a)+bar(b)| = |bar(a)-bar(b)|` then find the angle between `bar(a) and bar(b)`.

Text Solution

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Let `vec(a) , vec(b)` are the two vectors .
Sum of vectors
` = bar(a) + bar(b) = sqrt( a^(2) + b^(2) + 2ab cos theta)`
Difference of vectors
`= bar(a) - bar(b) = sqrt(a^(2) + b^(2) - 2ab cos theta)`
Given ` |bar(a) + bar(b) | = | bar(a) - bar(b)|`
` rArr sqrt(a^(2) + b^(2) + 2ab cos theta)`
` = sqrt(a^(2) + b^(2) - 2ab cos theta)`
by squaring on both sides ,
`a^(2) + b^(2) + 2ab cos theta = a^(2) + b^(2) - 2 ab cos theta`
` :.` 4 ab cos `theta = 0 ` or `theta = 90^(@)`
So if `|bar(a) + bar(b) | = |bar(a) - bar(b)|` then angle between `bar(a) ` and `bar(b) is 90^(@)` .
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