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WZ
Numerade Educator

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Problem 45 Medium Difficulty

Show that the vector $ orth_a b = b - proj_a b $ is orthogonal to $ a $. (It is called an orthogonal projection of $ b $.)

Answer

$\mathbf{a} \cdot$ ortho $\mathbf{b}=\mathbf{a} \cdot \mathbf{b}-\frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}|^{2}} \mathbf{a} \cdot \mathbf{a}$
$a \cdot$ ortho $_{\mathrm{a}} \mathrm{b}=\mathrm{a} \cdot \mathrm{b}-\mathrm{a} \cdot \mathrm{b}$
$\mathbf{a} \cdot \mathbf{b}=0 \quad$ (R.T.P)

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Video Transcript

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