Projections and Point-Line Distance

Projection and perpendicular distance

Signed scalar projection

\[\operatorname{comp}_{\mathbf a}\mathbf b=\mathbf b\cdot\widehat{\mathbf a}=\frac{\mathbf a\cdot\mathbf b}{|\mathbf a|}\qquad(\mathbf a\ne\mathbf0).\]

Projection length and vector projection

\[\text{length}=\left|\mathbf b\cdot\widehat{\mathbf a}\right|,\qquad \operatorname{proj}_{\mathbf a}\mathbf b=\frac{\mathbf a\cdot\mathbf b}{|\mathbf a|^2}\mathbf a.\]

Shortest distance to the line through O in direction a

\[d=\frac{|\mathbf a\times\mathbf b|}{|\mathbf a|}=|\mathbf b|\sin\theta\qquad(\mathbf a\ne\mathbf0).\]

A point $P$ with position vector $\mathbf p$ lies on $l:\mathbf r=\mathbf a+\lambda\mathbf d$ exactly when one scalar $\lambda$ satisfies every component of $\mathbf p=\mathbf a+\lambda\mathbf d$

Foot of the perpendicular

Foot and shortest distance

\[\mathbf n=\mathbf a+\lambda\mathbf d,\qquad (\mathbf p-\mathbf n)\cdot\mathbf d=0,\qquad PN=|\mathbf p-\mathbf n|.\]

Direct distance formula

\[\operatorname{dist}(P,l)=\frac{|(\mathbf p-\mathbf a)\times\mathbf d|}{|\mathbf d|}\qquad(\mathbf d\ne\mathbf0).\]

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Answer:Length of projection: $|\mathbf a\cdot\hat{\mathbf d}|=\dfrac{|\mathbf a\cdot\mathbf d|}{|\mathbf d|}\text{ units}$ Shortest distance: $|\mathbf a\times\hat{\mathbf d}|=\dfrac{|\mathbf a\times\mathbf d|}{|\mathbf d|}\text{ units}$

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