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Coplanar and Linear dependency.

by jrotmensen
Tags: coplanar, dependency, linear
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jrotmensen
#1
Dec26-09, 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)
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Dec26-09, 04:33 AM
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Quote Quote by jrotmensen View Post
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)
State the entire properties! What you have are equations, not properties.

"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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