Q. 9

Three vectors A, B and C add up to zero. Find which is false.
A. (A×B) × C is not zero unless B, C are parallel

B. (A×B).C is not zero unless B, C are parallel

C. If A, B, C define a plane, (A×B) ×C is in that plane

D. (A×B).C=|A||B||C| C2=A2+B2

Answer :

We have to identify false statement from above


It is given that


Therefore taking cross product on both side





Now we know that when vectors are parallel then their cross product is zero



Taking post Cross Product on both side with C



Now this could only zero when B and C are parallel to each other as,


only when that’s when B and C are parallel


Therefore statement A is true.


Now taking previous equation



Taking dot product with C on both side



Now this could be zero on two conditions first is that B and C are parallel but it could be zero without C being parallel to B. As when we will take cross product of B and C, then vector perpendicular to both B and C, say vector K. And by taking dot product of K and C it will also be zero as angle between them will always be 90.Therefore B is false


Now if vector triple product of A and B and C, then vector will always lie on place which will formed by A, B and C. This could be visualized by understanding that will always lie in a single plane forming sides of triangle.


Now,



K will be perpendicular to plane containing A and B.


And taking cross product with C (which is also lying on same plane as that of A and B) will give a vector which is perpendicular to C but will be lying on same plane as that of A,B and C. Therefore statement C is true.


It is given in last option that therefore angle between vector A and vector B is 90 and we know that


, therefore form a triangle with angle between A and B equal to 90, therefore it is right angled triangle. Hence option D is also true.

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