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