Why does a third vector lying on the plane spanned by two others result in linear dependence?

Conditions

• Three vectors exist in 3D space
• The first two vectors are linearly independent (non-parallel)
• The third vector lies within the plane spanned by the first two

Reasoning, step by step

1. Establish the plane formed by the first two independent vectors.
2. Check if the third vector's tip lies on this existing plane.
3. If yes, express the third vector as $c_1\vec{v}_1 + c_2\vec{v}_2$.
4. Recognize that adding this vector does not expand the reachable region (span) beyond the current plane.
5. Classify the set as linearly dependent due to redundancy.

Example

If $\vec{v}_1$ and $\vec{v}_2$ define the floor, and $\vec{v}_3$ is a shadow on the floor, $\vec{v}_3$ doesn't help you reach the ceiling. It is dependent on $\vec{v}_1$ and $\vec{v}_2$.

Common misconceptions

• Confusing linear dependence with orthogonality; dependent vectors don't need to be parallel, just coplanar in this context.
• Thinking that a zero vector is required for dependence; while true, non-zero coplanar vectors also exhibit dependence.

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Connected concepts

Explore next

• What characterizes a linearly independent set of three vectors?
• How does linear independence relate to the definition of a basis?

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