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Easy

Question

The pairs a with bar on top comma b with bar on top text  and  end text stack c with not stretchy bar on top comma d with space bar on top with blank on top each determines a plane. Then the planes are parallel if

  1. left parenthesis stack a with bar on top cross times stack c with bar on top right parenthesis cross times left parenthesis stack b with bar on top cross times stack d with bar on top right parenthesis equals stack 0 with bar on top    
  2. left parenthesis stack a with minus on top cross times stack c with minus on top right parenthesis times left parenthesis stack b with minus on top cross times stack d with minus on top right parenthesis equals 0    
  3. left parenthesis stack a with minus on top cross times stack b with minus on top right parenthesis cross times left parenthesis stack c with minus on top cross times stack d with minus on top right parenthesis equals stack 0 with bar on top    
  4. left parenthesis stack a with minus on top cross times stack b with minus on top right parenthesis times left parenthesis stack c with minus on top cross times stack d with minus on top right parenthesis equals 0    

hintHint:

We are given two pairs of the vectors. They both make planes. We have to find their relation when the planes are parallel to each other.

The correct answer is: left parenthesis stack a with minus on top cross times stack b with minus on top right parenthesis cross times left parenthesis stack c with minus on top cross times stack d with minus on top right parenthesis equals stack 0 with bar on top


    The given pairs of the planes are a with rightwards arrow on top comma b with rightwards arrow on top space a n d space c with rightwards arrow on top comma d with rightwards arrow on top

    When two vectors make up a plane, their cross product gives normal of the plane. Cross product is a vector product of the two vectors. 
    If two planes are parallel their normal will be parallel.
    The normal of the first plane is a with rightwards arrow on top cross times b with rightwards arrow on top
    The normal of the second plane is c with rightwards arrow on top cross times d with rightwards arrow on top
    When two planes are parallel their cross product is zero.
    So, open parentheses a with rightwards arrow on top cross times b with rightwards arrow on top close parentheses cross times open parentheses c with rightwards arrow on top cross times d with rightwards arrow on top close parentheses space equals space 0
    This is the required answer.

    Cross product of two vectors is equal to the product of their magnitudes and sin of the angle between them. When two vectors are parallel, the angle between them is zero resulting their dot product as zero.

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