Assertion ( A ) : If a ¯ ,   b ¯ are two non-collinear vectors then the vector component of b…

Assertion (A): If a¯, b¯ are two non-collinear vectors then the vector component of b¯ along the line perpendicular to a¯ is a¯×(b¯×a¯)|a¯|2.
Reason (R): a¯×(b¯×c¯)=(a¯·c¯)b¯-(a¯·b¯)c¯ and vector component of b¯ on c¯ is b¯·c¯|c¯|c¯|c¯|.
  1. Both (A) and (R) are true and (R) is the correct explanation of (A)
  2. Both (A) and (R) are true but (R) is not the correct explanation of (A)
  3. (A) is true but (R) is false
  4. (A) is false but (R) is true

Solution

The vector component b→ along the line parallel to a→ is

a→·b→|a→|a→|a→|

The figure below represents the vectors.

OA¯=a→·b→|a→|a→

Now,

AB→=OB→-OA→

=b→-a→·b→a→2a→

=(a→·a→)b→-(a→·b→)a→a→2

=a→×(b→×a→)a→2

Thus, (A) and (R) are true and (R) is the correct explanation.

Asked in: AP EAMCET 2019 (21 Apr Shift 2)

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