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Let  and  be three vectors such that   and   are co-planar. Which of the following is/are correct?

1.  is co-planar with  and 

2.  is perpendicular to  and 

Select the correct answer using the code given below.

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  1. 1 only
  2. 2 only
  3. Both 1 and 2
  4. Neither 1 nor 2

Answer (Detailed Solution Below)

Option 1 : 1 only
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Detailed Solution

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Concept:

1. If  and  be three vectors such that   and  are co-planar.

 = 0

2.Vector  is perpendicular to both vector   and .

Calculation:

We have,

  =  (Let)

⇒  is ⊥ to both  and .

But  is ⊥ to   and .

So,  or  will be coplanar to vector  and .

∴ Statement (1) is correct.

For statement 2:

Statement 2 is only possible when the vector  is parallel or anti-parallel to vector , which is not possible as per the above explanation for statement 1. So, this statement 2 is incorrect.

So, the best answer will be option 1.

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