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Chapter #2 :Vectors : Short Q/A / C.R.Q's

Q. Can a vector has zero magnitude if one of its components is not zero?
Ans:No A vector comprises its components, which are orthogonal. If just one of them has magnitude and direction, then the resultant vector has magnitude and direction. Example:-If A is a vector and Ax is zero and Ay is non-zero then, A=Ax+Ay  then A=0+Ay A=Ay

Q. Can the resultant of two vectors of the same magnitude  be  equal to the magnitude of either of the vectors? Give mathematical reasons for your answer.
Ans. Yes. If the angle between these two vectors is 120 degrees than it is possible.
For example:-  Consider two vector having magnitude A then
                        R=√A2+A2+2 A.A Cosθ         If R=A then    A=A√2Cos2θ/2
=> R=√2 A2+2 A2 Cosθ                     or 1=√2Cos2θ/2  
                        R=√2 A2    (1+ Cosθ)               S.O.B.S
=> R=A√2Cos2θ/2                  1/2= Cos2θ /2 then θ /2 =60    
                                                                        Or θ=120 Degrees
Q. Can two vectors having different magnitudes be combined to give zero resultant? Can three vectors?
Ans. The Resultant of two vectors having different magnitudes can not equal to zero,If they have same magnitude and opposite direction then their resultant will be equal to zero.
            Where as the resultant of three vectors equal to zero if they form triangle after combination.
Then A+B+C=0 
Q. How is it possible that i,j and k are vectors but don’t have magnitude?
Ans. The Unit Vectors (i , j , k ) are used to show the direction of any vector and they don’t represent any physical quantity therefore they have no units.

Q. For Dot Product A.B=AB cosθ. Why cosθ?
Ans. The Dot product is “The product of magnitude of any vector to the magnitude of projection of  another vector”.In the mathematical formula The cosθ  represents the projection of one vector over another along X-axis.

Q. If A x B =AB and A.B=0 then what is the angle?
Ans.     A X B =AB Sinθ                                A . B =AB Cosθ
            If θ= 90 degrees                                              If θ= 90 degrees         
Then    A X B= AB    (b/c Sin90=1)              Then A.B =0   (b/c Cos 90=0)

Q. Can the magnitude of any vector will be negative?
Ans. No the magnitude of any vector can not be negative because magnitude is calculated by
            A=√ X2 +Y2 +Z2
            So after squaring, if there is any negative sign it will be removed. hats why magnitude will not be negative


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