# Prove That Linear Combination is Coplanar

In summary, using the triple scalar product, it can be shown 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 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?

Last edited:

## 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

## 1. What is a linear combination?

A linear combination is a mathematical operation that involves multiplying each element in a set of numbers by a constant and adding them together. For example, if we have the numbers 2, 3, and 4, and we multiply 2 by a constant of 3 and add it to 3 multiplied by a constant of 5 and add it to 4 multiplied by a constant of 2, the result would be 26 (2x3 + 3x5 + 4x2 = 26).

## 2. What does it mean for a linear combination to be coplanar?

When a linear combination is coplanar, it means that all of the vectors involved in the combination lie in the same plane in three-dimensional space. This can also be thought of as all of the vectors being parallel to each other.

## 3. How do you prove that a linear combination is coplanar?

In order to prove that a linear combination is coplanar, you must show that all of the vectors involved in the combination can be written as a linear combination of two other vectors. If this is possible, then the vectors are parallel and therefore coplanar.

## 4. Can a linear combination of more than three vectors be coplanar?

Yes, a linear combination of any number of vectors can be coplanar as long as they can all be written as a linear combination of two other vectors. However, the more vectors that are involved, the more difficult it may be to prove that they are coplanar.

## 5. What is the significance of proving that a linear combination is coplanar?

Proving that a linear combination is coplanar can be useful in various mathematical and scientific applications. For example, it can help in determining if a set of vectors in a three-dimensional space can be reduced to two dimensions without losing any important information. It can also aid in solving systems of linear equations and understanding the relationships between vectors in a given system.