Prove That Linear Combination is Coplanar

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Homework Statement


Show that if [itex]c=\alpha{a}+\beta{b}[/itex], where [itex]a[/itex] and [itex]b[/itex] are arbitrary vectors and [itex]\alpha[/itex] and [itex]\beta[/itex] are arbitrary scalars, then [itex]c[/itex] is coplanar with [itex]a[/itex] and [itex]b[/itex].

Homework Equations


Triple scalar product: [itex](a\cdot{b})\times{c}=0[/itex]

The Attempt at a Solution


[itex]0=a\times(\alpha{a}+\beta{b})\cdot{b}[/itex]
[itex]0=(\alpha{a\times{a}}+\beta{a\times{b}})\cdot{b}[/itex]
[itex]0=\beta({a\times{b}})\cdot{b}[/itex]
[itex]0=({a\times{b}})\cdot{b}[/itex]
[itex]0=({b\times{b}})\cdot{a}[/itex]

Is this right?
 
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Answers and Replies

  • #2
ehild
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Homework Statement


Show that if [itex]c=\alpha{a}+\beta{b}[/itex], where [itex]a[/itex] and [itex]b[/itex] are arbitrary vectors and [itex]\alpha[/itex] and [itex]\beta[/itex] are arbitrary scalars, then [itex]c[/itex] is coplanar with [itex]a[/itex] and [itex]b[/itex].

Homework Equations


Triple scalar product: [itex](a\cdot{b})\times{c}=0[/itex]

The Attempt at a Solution


[itex]0=a\times(\alpha{a}+\beta{b})\cdot{b}[/itex]
[itex]0=(\alpha{a\times{a}}+\beta{a\times{b}})\cdot{b}[/itex]
[itex]0=\beta({a\times{b}})\cdot{b}[/itex]
[itex]0=({a\times{b}})\cdot{b}[/itex]
[itex]0=({b\times{b}})\cdot{a}[/itex]

Is this right?

(a˙b) is a scalar, multiplied by a vector is zero only when either the scalar or the vector is zero. Correctly, the triple scalar product is

[itex]\vec a \cdot (\vec b\times\vec c)=\vec b \cdot (\vec c\times\vec a)=\vec c \cdot (\vec a \times \vec b)[/itex].

Take care of the parentheses, it will be all right. The method is good.


ehild
 

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