Prove That Linear Combination is Coplanar

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Homework Statement

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

Homework Equations

Triple scalar product: $(a\cdot{b})\times{c}=0$

The Attempt at a Solution

$0=a\times(\alpha{a}+\beta{b})\cdot{b}$
$0=(\alpha{a\times{a}}+\beta{a\times{b}})\cdot{b}$
$0=\beta({a\times{b}})\cdot{b}$
$0=({a\times{b}})\cdot{b}$
$0=({b\times{b}})\cdot{a}$

Is this right?

 

[ehild]

Homework Statement

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

Homework Equations

Triple scalar product: $(a\cdot{b})\times{c}=0$

The Attempt at a Solution

$0=a\times(\alpha{a}+\beta{b})\cdot{b}$
$0=(\alpha{a\times{a}}+\beta{a\times{b}})\cdot{b}$
$0=\beta({a\times{b}})\cdot{b}$
$0=({a\times{b}})\cdot{b}$
$0=({b\times{b}})\cdot{a}$

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

$\vec a \cdot (\vec b\times\vec c)=\vec b \cdot (\vec c\times\vec a)=\vec c \cdot (\vec a \times \vec b)$.

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


ehild