
#1
Dec2609, 01:12 AM

P: 3

1. The problem statement, all variables and given/known data
Prove that vectors u, v, w are coplanar if and only if vectors u, v and w are linearly dependent. [tex]\overline{v}_{3}=\alpha\overline{v}_{1}+\beta\overline{v}_{2}[/tex] (Coplanar Vector Property) [tex]\alpha\overline{v}_{1}+\beta\overline{v}_{2}+\gamma\overline{v}_{3}=\ov erline{0}[/tex] (linearly dependent vector property) 



#2
Dec2609, 04:33 AM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 38,881

"Three vectors, [itex]v_1[/itex], [itex]v_2[/itex], and [itex]v_3[/itex] are coplanar if and only if [tex]\overline{v}_{3}=\alpha\overline{v}_{1}+\beta\overline{v}_{2}[/tex] or [tex]\overline{v}_{1}=\alpha\overline{v}_{2}+\beta\overline{v}_{3}[/tex] or [tex]\overline{v}_{2}=\alpha\overline{v}_{1}+\beta\overline{v}_{3}[/tex] for some numbers [itex]\alpha[/itex] and [itex]\beta[/itex]" "Three vectors, [itex]v_1[/itex], [itex]v_2[/itex], and [itex]v_3[/itex] are dependent if [tex]\alpha\overline{v}_{1}+\beta\overline{v}_{2}+\gamma\overline{v}_{3}=\ov erline{0}[/tex] with not all of [itex]\alpha[/itex], [itex]\beta[/itex], [itex]\gamma[/itex] equal to 0." Suppose [itex]\vec{v_1}[/itex], [itex]\vec{v_2}[/itex], and [itex]\vec{v_3}[/itex] are planar. Subtract the right side of that equation from both sides. Suppose [itex]\vec{v_1}[/itex], [itex]\vec{v_2}[/itex], and [itex]\vec{v_3}[/itex] are dependent. Solve that equation for one of the vectors. 


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