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.
  • #1
15
0

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?
 
Last edited:
Physics news on Phys.org
  • #2
blink- said:

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
 

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.

Suggested for: Prove That Linear Combination is Coplanar

Replies
1
Views
325
Replies
8
Views
1K
Replies
2
Views
798
Replies
7
Views
705
Replies
57
Views
2K
Replies
4
Views
829
Replies
21
Views
1K
Replies
1
Views
724
Back
Top