What-condition-should-be-satisfied-by-the-vectors-a-and-b-for-the-following-relations-to-hold-true-a-a-b-a-b-b-a-b-gt-a-b-c-a-b-lt-a-b-

Question Number 118286 by 1549442205PVT last updated on 16/Oct/20

What condition should be satisfied by

the vectors \boldsymbol a and \boldsymbol b for the following

relations to hold true : (a) ∣ \boldsymbol a + \boldsymbol b ∣=∣ \boldsymbol a - \boldsymbol b ∣

; (b) ∣ \boldsymbol a + \boldsymbol b ∣>∣ \boldsymbol a - \boldsymbol b ∣; (c) ∣ \boldsymbol a + \boldsymbol b ∣< ∣ \boldsymbol a - \boldsymbol b ∣

Answered by TANMAY PANACEA last updated on 16/Oct/20

a) ∣ \boldsymbol a + \boldsymbol b ∣=∣ \boldsymbol a - \boldsymbol b ∣

∣ \boldsymbol a + \boldsymbol b ∣^{2} =∣ \boldsymbol a - \boldsymbol b ∣^{2}

(\boldsymbol a + \boldsymbol b) . (\boldsymbol a + \boldsymbol b) = (\boldsymbol a - \boldsymbol b) . (\boldsymbol a - \boldsymbol b)

a^{2} + b^{2} + 2 a b c o s θ = a^{2} + b^{2} - 2 a b c o s θ

[\boldsymbol a . \boldsymbol b = a b c o s θ \boldsymbol a . \boldsymbol a = a^{2} \boldsymbol b . \boldsymbol b = b^{2}]

4 a b c o s θ = 0 θ = \frac{π}{2} s o \boldsymbol a ⊥ \boldsymbol b

Answered by TANMAY PANACEA last updated on 16/Oct/20

b) s i m i l a r l y

∣ \boldsymbol a + \boldsymbol b ∣>∣ \boldsymbol a - \boldsymbol b ∣

\sqrt{a^{2} + b^{2} + 2 a b c o s θ} > \sqrt{a^{2} + b^{2} - 2 a b c o s θ}

c o s θ > 0 s o θ = a c u t e a n g l e

\frac{π}{2} > θ > 0

c) \boldsymbol a + \boldsymbol b ∣<∣ \boldsymbol a - \boldsymbol b ∣

\sqrt{a^{2} + b^{2} + 2 a b c o s θ} < \sqrt{a^{2} + b^{2} - 2 a b c o s θ}

c o s θ = - v e

c o s θ < 0

π > θ > \frac{π}{2} c o r r e c t i o n d o n e

p l z c h e c k

Commented by TANMAY PANACEA last updated on 16/Oct/20

m o s t w e l c o m e s i r

Commented by TANMAY PANACEA last updated on 16/Oct/20

y e s s i r \dots

Commented by 1549442205PVT last updated on 16/Oct/20

Thank sir . You are welcome .

You corrected exactly . When

θ = π then there two possibilities

∣ a + b ∣=∣ a - b ∣ = 0 o r ∣ a + b ∣<∣ a - b ∣

Question means the inequalities are

really true .