Question

If a and b are proper vectors, show that la+bl=la-b| if and only if , and b are orthogonal.​

Answer 1

answer:--

Let us suppose the angle (defined from a to b clockwise) between the vectors is θ,0≤θ≤2π . Then using the parallelogram rule for vector addition, and the law of cosines

|a+b|2=|a|2+|b|2−2|a||b|cosθ.

Similarly, recognizing that the angle between a and −b is π−θ , we get

|a−b|2=|a|2+|b|2−2|a||b|cos(π−θ)=|a|2+|b|2+2|a||b|cosθ.

As, |a+b|=|a−b| ,

|a|2+|b|2−2|a||b|cosθ=|a|2+|b|2+2|a||b|cosθ.

Or, on simplification, cosθ=0 .

Hence, θ=π2 or 3π2 .

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