Question

for two vectors a and b if vector R=vector a+vector b and vector S=vector a-vector b if 2|vectorR|=|vector S|,when vector R is perpendicular to vector a,then-​

Answer 1

Given info : for two vector a and b if R = a + b and S = a - b if 2|R| = |S| and R is perpendicular to a.

solution : let Ф is angle between vectors a and b.

R = a + b

∵ R is perpendicular to a.

∴ R.a = 0

so, R.a = (a + b).a

⇒ 0 = a.a + a.b

⇒ 0 = a² + abcosФ

⇒ cosФ = -a/b ..(1)

given, 2|R| = |S|

⇒ 4|R|² = |S|²

⇒ 4(a² + b² + 2abcosФ) = (a² + b² - 2abcosФ)

⇒ 4(a² + b²-2a²) = (a² + b² + 2a²)                 [ from eq (1) ]

⇒ 4b² - 4a² = 3a² + b²

⇒ 3b² = 7a²

⇒ √3 b = ± √7 a

therefore the relation between a and b will be √3 b = ± √7 a