Question
True or False For any two nonzero vectors $\mathbf{v}$ and $\mathbf{w},$ the vector $\mathbf{v}$ can be decomposed into two vectors, one parallel to $\mathbf{w}$ and the other orthogonal to $\mathbf{w}$.
Step 1
We can do this by projecting $\mathbf{v}$ onto $\mathbf{w}$. The projection of $\mathbf{v}$ onto $\mathbf{w}$ is given by the formula: \[\mathbf{v}_{\parallel} = \frac{\mathbf{v} \cdot \mathbf{w}}{\mathbf{w} \cdot \mathbf{w}} \mathbf{w}\] where $\mathbf{v} \cdot Show more…
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