Question

Two vectors A and B are given such that |A| = |B| then what is the angle between |A+B| and |A-B|

Answer 1

Answer:

(A+B).(A−B)=|A+B|∗|A−B|∗cosϕ

=>|A|2+A.B−A.B−|B|2=(|A|2+|B|2+2ABcosθ−−−−−−−−−−−−−−−−−−√)(|A|2+|B|2+2ABcosθ−−−−−−−−−−−−−−−−−−√)∗cosϕ

=>|A|2−|B|2=(|A|2+|B|2+2ABcosθ)(|A|2+|B|2−2ABcosθ)−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√∗cosϕ

=>|A|2−|B|2=(|A|2+|B|2)2−(2ABcosθ)2−−−−−−−−−−−−−−−−−−−−−−−√∗cosϕ

=>|A|2−|B|2=|A|4+|B|4+2|A|2|B|2–4|A|2|B|2cos2θ−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√∗cosϕ

=>|A|2−|B|2=|A|4+|B|4+2|A|2|B|2(1–2cos2θ)−−−−−−−−−−−−−−−−−−−−−−−−−−−√∗cosϕ

=>|A|2−|B|2=|A|4+|B|4+2|A|2|B|2(−cos2θ)−−−−−−−−−−−−−−−−−−−−−−−−−−√∗cosϕ

=>|A|2−|B|2=|A|4+|B|4−2|A|2|B|2cos2θ−−−−−−−−−−−−−−−−−−−−−−−√∗cosϕ

=>cosϕ=|A|2−|B|2|A|4+|B|4−2|A|2|B|2cos2θ

Answer 2

Explanation:

If a is the angle between (A+B) and A, then the angle between (A-B) and A will be 90 -a. Therefore, the angle between (A+B) and(A- B) is a+ 90 -a = 90 degree.

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