Coplanar and Linear dependency.

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In summary, three vectors, v_1, v_2, and v_3, are coplanar if and only if they can be expressed as multiples of each other, or if one vector can be expressed as a combination of the other two. They are dependent if one of the vectors can be written as a linear combination of the other two.
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Homework Statement


Prove that vectors u, v, w are coplanar if and only if vectors u, v and w are linearly dependent.[tex]\overline{v}_{3}=\alpha\overline{v}_{1}+\beta\overline{v}_{2}[/tex] (Coplanar Vector Property)
[tex]\alpha\overline{v}_{1}+\beta\overline{v}_{2}+\gamma\overline{v}_{3}=\overline{0}[/tex] (linearly dependent vector property)
 
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jrotmensen said:

Homework Statement


Prove that vectors u, v, w are coplanar if and only if vectors u, v and w are linearly dependent.


[tex]\overline{v}_{3}=\alpha\overline{v}_{1}+\beta\overline{v}_{2}[/tex] (Coplanar Vector Property)
[tex]\alpha\overline{v}_{1}+\beta\overline{v}_{2}+\gamma\overline{v}_{3}=\overline{0}[/tex] (linearly dependent vector property)
State the entire properties! What you have are equations, not properties.

"Three vectors, [itex]v_1[/itex], [itex]v_2[/itex], and [itex]v_3[/itex] are coplanar if and only if
[tex]\overline{v}_{3}=\alpha\overline{v}_{1}+\beta\overline{v}_{2}[/tex]
or
[tex]\overline{v}_{1}=\alpha\overline{v}_{2}+\beta\overline{v}_{3}[/tex]
or
[tex]\overline{v}_{2}=\alpha\overline{v}_{1}+\beta\overline{v}_{3}[/tex]
for some numbers [itex]\alpha[/itex] and [itex]\beta[/itex]"

"Three vectors, [itex]v_1[/itex], [itex]v_2[/itex], and [itex]v_3[/itex] are dependent if [tex]\alpha\overline{v}_{1}+\beta\overline{v}_{2}+\gamma\overline{v}_{3}=\overline{0}[/tex]
with not all of [itex]\alpha[/itex], [itex]\beta[/itex], [itex]\gamma[/itex] equal to 0."

Suppose [itex]\vec{v_1}[/itex], [itex]\vec{v_2}[/itex], and [itex]\vec{v_3}[/itex] are planar. Subtract the right side of that equation from both sides.

Suppose [itex]\vec{v_1}[/itex], [itex]\vec{v_2}[/itex], and [itex]\vec{v_3}[/itex] are dependent. Solve that equation for one of the vectors.
 
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Related to Coplanar and Linear dependency.

1. What is the difference between coplanar and linear dependency?

Coplanar describes a set of points, vectors, or lines that all lie in the same plane, while linear dependency refers to a set of vectors that can be written as a linear combination of each other. In other words, coplanar means all the points are on the same plane, while linear dependency means the vectors are dependent on each other.

2. Why is it important to understand coplanar and linear dependency in math and science?

Coplanar and linear dependency are important concepts in math and science because they help us understand and analyze systems of points, vectors, and lines in three-dimensional space. They also have practical applications in fields such as physics, engineering, and computer graphics.

3. How can you tell if a set of points is coplanar?

A set of points is coplanar if they can all be contained in the same plane. This can be determined by visually inspecting the points or by using mathematical tests, such as checking if the points satisfy a linear equation in three variables.

4. What is the significance of linearly dependent vectors?

Linearly dependent vectors have a special relationship with each other, where one or more vectors can be expressed as a linear combination of the others. This means that they lie on the same line or plane and do not add any new information to the system. In mathematics, linearly dependent vectors can lead to inconsistencies and are therefore important to identify and avoid.

5. How are coplanar and linearly dependent vectors related?

Coplanar vectors are always linearly dependent, but linearly dependent vectors are not always coplanar. This is because coplanar vectors must lie in the same plane, but linearly dependent vectors can still be contained in different planes or lie on different lines.

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