Prove that vectors u,v and w are coplanar if and only if vectors u, v and w are linearly dependent?
T Tess213 New member Joined Apr 18, 2008 Messages 1 Apr 18, 2008 #1 Prove that vectors u,v and w are coplanar if and only if vectors u, v and w are linearly dependent?
S soroban Elite Member Joined Jan 28, 2005 Messages 5,586 Apr 18, 2008 #2 Re: Vector Calculus.....PLEASE HELP Hello, Tess213! I have part of it ... well, sort of . . . Prove that vectors u⃗, v⃗ and w⃗ are coplanar\displaystyle \text{Prove that vectors }\vec{u},\:\vec{v}\text{ and }\vec{w}\text{ are coplanar}Prove that vectors u,v and w are coplanar . . if and only if vectors u⃗, v⃗ and w⃗ are linearly dependent.\displaystyle \text{ if and only if vectors }\vec{u},\;\vec{v}\text{ and }\vec{w}\text{ are linearly dependent.} if and only if vectors u,v and w are linearly dependent. Click to expand... If the vectors are linearly dependent, one vector is a linear combination of the other two. . . That is: .u⃗ = av⃗+bw⃗\displaystyle \vec{u} \:=\:a\vec{v} + b\vec{w}u=av+bw Then u⃗\displaystyle \vec{u}u is a "scalar sum" of the v⃗ and w⃗\displaystyle \vec{v}\text{ and }\vec{w}v and w Code: * * * * v * * u * * * * * * * * * w Hence, they are coplanar.
Re: Vector Calculus.....PLEASE HELP Hello, Tess213! I have part of it ... well, sort of . . . Prove that vectors u⃗, v⃗ and w⃗ are coplanar\displaystyle \text{Prove that vectors }\vec{u},\:\vec{v}\text{ and }\vec{w}\text{ are coplanar}Prove that vectors u,v and w are coplanar . . if and only if vectors u⃗, v⃗ and w⃗ are linearly dependent.\displaystyle \text{ if and only if vectors }\vec{u},\;\vec{v}\text{ and }\vec{w}\text{ are linearly dependent.} if and only if vectors u,v and w are linearly dependent. Click to expand... If the vectors are linearly dependent, one vector is a linear combination of the other two. . . That is: .u⃗ = av⃗+bw⃗\displaystyle \vec{u} \:=\:a\vec{v} + b\vec{w}u=av+bw Then u⃗\displaystyle \vec{u}u is a "scalar sum" of the v⃗ and w⃗\displaystyle \vec{v}\text{ and }\vec{w}v and w Code: * * * * v * * u * * * * * * * * * w Hence, they are coplanar.
