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Question

If a with minus on top equals i with minus on top plus j with minus on top plus k with minus on top text  and  end text b with minus on top equals stack i with minus on top minus j with minus on top with blank on top then the vectors Error converting from MathML to accessible text. and Error converting from MathML to accessible text.

  1. Are mutually Perpendicular    
  2. Coplanar    
  3. Forms a Parallelopiped of volume 2 units    
  4. Forms a Parallelopiped of volume 3 units    

Hint:

We have three vectors. Two of the vectors are based on the other two given vectors. We have to find the relation between the vectors.

The correct answer is: Are mutually Perpendicular


    The vectors a with rightwards arrow on top a n d b with rightwards arrow on top are given as follows:
    table attributes columnalign left end attributes row cell a with rightwards arrow on top equals i with hat on top plus j with hat on top plus k with hat on top end cell row cell b with rightwards arrow on top equals i with hat on top minus j with hat on top end cell end table
    Let the other three vectors be denoted as P with rightwards arrow on top comma stack Q space with rightwards arrow on top a n d space R with rightwards arrow on top
    P with rightwards arrow on top equals open parentheses a with rightwards arrow on top.1 with overparenthesis on top close parentheses i with overparenthesis on top plus open parentheses a with rightwards arrow on top. j with overparenthesis on top close parentheses i with overparenthesis on top plus open parentheses a with rightwards arrow on top. k with overparenthesis on top close parentheses k with hat on top
    Q with rightwards arrow on top equals open parentheses b with rightwards arrow on top. i with hat on top close parentheses i with hat on top plus open parentheses b with rightwards arrow on top. j with hat on top close parentheses j with hat on top plus open parentheses b with rightwards arrow on top. k with hat on top close parentheses k with hat on top
    R with rightwards arrow on top equals i with hat on top plus j with hat on top minus 2 k with overparenthesis on top
    We have to find the relation between the vectors.
    Now, we will first find the components of the vectors
    a with rightwards arrow on top. i with overparenthesis on top equals left parenthesis i with hat on top plus i with hat on top plus k with hat on top right parenthesis open parentheses i with hat on top close parentheses equals 1
stack a. with rightwards arrow on top j with hat on top equals left parenthesis i with hat on top plus j with hat on top plus k with hat on top right parenthesis left parenthesis j with hat on top right parenthesis space equals space 1
a with rightwards arrow on top. k with hat on top equals left parenthesis i with hat on top plus j with hat on top plus k with hat on top right parenthesis k with hat on top equals 1
    b with rightwards arrow on top. i with overparenthesis on top equals open parentheses i with hat on top minus j with hat on top close parentheses open parentheses i with hat on top close parentheses equals 1
b with rightwards arrow on top. j with hat on top space equals left parenthesis i with hat on top minus j with hat on top right parenthesis left parenthesis j with hat on top right parenthesis space equals space minus 1
b with rightwards arrow on top. k with hat on top equals left parenthesis i with hat on top minus j with hat on top right parenthesis left parenthesis k with hat on top right parenthesis space equals space 0
    So, the vectors become.
    P with rightwards arrow on top equals i with hat on top plus j with hat on top plus k with hat on top
Q with rightwards arrow on top equals i with hat on top minus j with hat on top
    Now, we will take their dot product to check if they are mutually perpendicular or not.
    P with bar on top. Q with bar on top equals open parentheses i with hat on top plus j with hat on top plus k with hat on top close parentheses. open parentheses i with hat on top minus j with hat on top close parentheses
space space space space space space space space equals space 1 space minus space 1
space space space space space space space space space equals space 0
S o comma space t h e space g i v e n space v e c t o r s space a r e space p e r p e n d i c l u a r.
    Q with rightwards arrow on top. R with rightwards arrow on top equals open parentheses i with hat on top minus j with hat on top close parentheses open parentheses i with hat on top plus j with hat on top minus 2 k with hat on top close parentheses
space space space space space space space space equals space 1 minus 1
space space space space space space space space space equals space 0
S o comma space t h e space g i v e n space v e c t o r s space a r e space p e r p e n d i c u l a r

P with rightwards arrow on top. R with rightwards arrow on top equals open parentheses i with hat on top plus j with hat on top plus k with hat on top close parentheses left parenthesis i with hat on top plus j with hat on top minus 2 k with hat on top right parenthesis
space space space space space space space space equals space 1 space plus space 1 space minus space 2
space space space space space space space space equals space 0

space space space space space space space space
    So, the given vectors are perpendicular.
    If we see all the vectors are perpendicular to each other.
    The given vectors are mutually perpendicular.

    We can check for parallelepiped. We can take the scalar product of the given vectors. If we do that, we will find the volume to be 6 units. It doesn't match any option.

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