Let x → ,   y → and z → be three vectors each of magnitude 2 and the angle between…

Let x→, y→ and z→ be three vectors each of magnitude 2 and the angle between each pair of them is π3. If a→ is a non - zero vector perpendicular to x→ and y→ ×z→ and b→ is a non - zero vector perpendicular to y→ and z→ ×x→ , then
  1. b→=b→ . z→ z→- x→
  2. a→=a→ . y→ y→- z→
  3. a→ . b→= -a→ . y→ b→ . z→
  4. a→=a→ . y→ z→- y→

Solution

a→ is in direction of x→ ×y→ × z→
i.e., x→ . z→ y→-x→ . y→ z→
⇒ a→=λ12 ×12 y→- z→
a→=λ1y→- z→ ........(i)
Now a→ . y→=λ1 y→ . y→- y→ . z→
=λ1 2-1⇒  λ1= a→ . y→ ........(ii)
From (i) and (ii), a→= a→ . y→ y→- z→
Similarly, b→=b→ . z→ z→-x→
Now, a→ . b→=a→ . y→ b→ . z→ y→- z→ . z→- x→
=a→ . y→ b→ . z→ 1-1-2+1
= -a→ . y→ b→ . z→

Asked in: JEE Advanced 2014 (Paper 1)

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