If a vector vecα lie in the plane of vecβ and vecγ then which is correct?

Question

If a vector vecα lie in the plane of vecβ and vec&gamma; then which is correct? (A) [ vecα vecβ vec&gamma;]=0 (B) [ vecα vecβ vec&gamma;]=1 (C) [ vecα vecβ vec&gamma;]=3 (D) [ vecβ vec&gamma; vecα ]=1

Tardigrade Admin · Accepted Answer

If vector vecα lie in the plane of vecβ and vec&gamma; then all three vectors vecα, vecβ and vec&gamma; are coplanar. For three vectors to be coplanar, their scalar triple product must be zero i.e. [ vecα vecβ vec&gamma; ]=0

Q. If a vector $α$ lie in the plane of $β$ and $γ$ then which is correct?

$[α β γ] = 0$

$[α β γ] = 1$

$[α β γ] = 3$

$[β γ α] = 1$

If vector $α$ lie in the plane of $β$ and $γ$ then all three vectors $α, β$ and $γ$ are coplanar.
For three vectors to be coplanar, their scalar triple product must be zero
i.e. $[α β γ] = 0$

Q. If a vector α lie in the plane of β​ and γ​ then which is correct?

[αβ​γ​]=0

[αβ​γ​]=1

[αβ​γ​]=3

[β​γ​α]=1