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A-B-and-C-are-three-non-collinear-non-co-planar-vectors-What-can-you-say-about-direction-of-A-B-C-




Question Number 18094 by Tinkutara last updated on 15/Jul/17
A^→ , B^(→)  and C^(→)  are three non-collinear,  non co-planar vectors. What can you  say about direction of A^(→) ×(B^(→) ×C^(→) )?
$$\overset{\rightarrow} {{A}},\:\overset{\rightarrow} {{B}}\:\mathrm{and}\:\overset{\rightarrow} {{C}}\:\mathrm{are}\:\mathrm{three}\:\mathrm{non}-\mathrm{collinear}, \\ $$$$\mathrm{non}\:\mathrm{co}-\mathrm{planar}\:\mathrm{vectors}.\:\mathrm{What}\:\mathrm{can}\:\mathrm{you} \\ $$$$\mathrm{say}\:\mathrm{about}\:\mathrm{direction}\:\mathrm{of}\:\overset{\rightarrow} {{A}}×\left(\overset{\rightarrow} {{B}}×\overset{\rightarrow} {{C}}\right)? \\ $$
Commented by ajfour last updated on 15/Jul/17
A^→ ×(B^→ ×C^→ ) is the cross product of   the component of A^→  in the plane  of B^→  and C^→  with B^→ ×C^→  .  It is parallel to the plane of B^→  and C^→ .
$$\overset{\rightarrow} {\mathrm{A}}×\left(\overset{\rightarrow} {\mathrm{B}}×\overset{\rightarrow} {\mathrm{C}}\right)\:\mathrm{is}\:\mathrm{the}\:\mathrm{cross}\:\mathrm{product}\:\mathrm{of} \\ $$$$\:\mathrm{the}\:\mathrm{component}\:\mathrm{of}\:\overset{\rightarrow} {\mathrm{A}}\:\mathrm{in}\:\mathrm{the}\:\mathrm{plane} \\ $$$$\mathrm{of}\:\overset{\rightarrow} {\mathrm{B}}\:\mathrm{and}\:\overset{\rightarrow} {\mathrm{C}}\:\mathrm{with}\:\overset{\rightarrow} {\mathrm{B}}×\overset{\rightarrow} {\mathrm{C}}\:. \\ $$$$\mathrm{It}\:\mathrm{is}\:\mathrm{parallel}\:\mathrm{to}\:\mathrm{the}\:\mathrm{plane}\:\mathrm{of}\:\overset{\rightarrow} {\mathrm{B}}\:\mathrm{and}\:\overset{\rightarrow} {\mathrm{C}}. \\ $$
Commented by ajfour last updated on 15/Jul/17
Commented by Tinkutara last updated on 15/Jul/17
Thanks Sir!
$$\mathrm{Thanks}\:\mathrm{Sir}! \\ $$

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