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-

Question Number 19976 by Tinkutara last updated on 19/Aug/17

Three vectors \vec{A}, \vec{B} and \vec{C} add up to

zero . Find which is false .

(a) (\vec{A} \times \vec{B}) \times \vec{C} is not zero unless \vec{B}, \vec{C}

are parallel

(b) (\vec{A} \times \vec{B}) \cdot \vec{C} is not zero unless \vec{B}, \vec{C}

are parallel

(c) If \vec{A}, \vec{B}, \vec{C} define a plane, (\vec{A} \times \vec{B} \times \vec{C})

is in that plane

(d) (\vec{A} \times \vec{B}) . \vec{C} = ∣ \vec{A} ∣∣ \vec{B} ∣∣ \vec{C} ∣ \to C^{2} = A^{2} + B^{2}

Answered by ajfour last updated on 19/Aug/17

(a), (b), (d) a r e f a l s e . f o r (\vec{A} \times \vec{B}) . \vec{C} t o b e

z e r o \vec{C} j u s t n e e d t o b e i n t h e p l a n e

o f \vec{A} a n d \vec{B} . F o r (\vec{A} \times \vec{B}) \times \vec{C} t o b e

z e r o \vec{C} n e e d t o b e p a r a l l e l t o \vec{A} \times \vec{B} .

I f (\vec{A} \times \vec{B}) . \vec{C} =∣ \vec{A} ∣∣ \vec{B} ∣∣ \vec{C} ∣, \vec{A}, \vec{B}, \vec{C}

j u s t n e e d b e m u t u a l l y p e r p e n d i c u l a r .

Commented by ajfour last updated on 19/Aug/17

(\hat{i} \times \hat{j}) \times \hat{j} = - \hat{i} . (a) i s f a l s e .

Commented by Tinkutara last updated on 20/Aug/17

Thank you very much Sir!