I'm trying to understand why if a vector is orthogonal to one vector in a plane, why it wouldn't be orthogonal to all vectors in that plane?
Sketches/diagrams would be helpful.
Vector that is orthogonal to one vector in a plane, automatically the normal?
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orthogonality
3 Answers
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Hint: Consider two vectors in the same plane that are orthogonal.
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Because given a vector $u$ in the plane $\alpha$, a vector $v$ can be orthogonal to $u$ and stay in the same plane $\alpha$. As an example: $$ \vec u=(1,0,0)^T \qquad \vec v=(0,1,0)^T$ $$ are orthogonal but stay in the same plane $x-y$.
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Think of the $xy$-plane in $\mathbb{R}^3$ and the vector $\hat{\text{i}}$. $\hat{\text{j}}$ is orthogonal to $\hat{\text{i}}$ but it's not orthogonal/normal to the plane.