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  • 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

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=\left | A \right |\left | B \right |\left | C \right |\rightarrow C^{2}=A^{2}+B^{2}

 

Answers (1)

The answer is the option (a) and (c)

Explanation:

Given \vec{A}+\vec{B}+\vec{C}=0

Option a : \vec{B}\times (\vec{A}+\vec{B}+\vec{C})=\vec{B}\times 0=0

0=\vec{B}\times \vec{A}+\vec{B}\times \vec{B}+\vec{B}\times \vec{C}

0=\vec{B}\times \vec{A}+0+\vec{B}\times \vec{C}

\vec{A}\times \vec{B}=\vec{B}\times \vec{C}(this cannot be zero)

Only if B and C are antiparallel or parallel  \vec{B}\times \vec{C} will be zero

Hence for the whole quantity to be zero, \vec{B}\parallel \vec{C} should be true.

Option C : \vec{A}\times \vec{B}=\vec{X}

X is perpendicular to the planes which have vector A and vector B

Vector Y is perpendicular to the planes which have vector A and vector B

Vector Y is perpendicular to the plane containing X and C which is, in turn, the plane containing vectors A, B and C

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