Vector playground

<p>If&nbsp;\(P\) divides \(AB\) internally with \(AP:PB=m:n\), then&nbsp;\(\overrightarrow{AP}=\dfrac{m}{m+n}\overrightarrow{AB}\). Starting at \(A\),</p>
<p>\(\overrightarrow{OP}=\mathbf{a}+\dfrac{m}{m+n}(\mathbf{b}-\mathbf{a})\).</p>
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<p>The fraction uses the part travelled from the starting point, not the remaining part. For a midpoint use \(\dfrac12\). A multiplier greater than 1 continues beyond \(B\); a negative multiplier travels in the opposite direction.</p>
<p>In a proof, derive a midpoint or ratio from the givens before using it. For area ratios, triangles with the same perpendicular height have areas in the ratio of their bases; no squared scale factor is needed.</p>

If \(P\) divides \(AB\) internally with \(AP:PB=m:n\), then \(\overrightarrow{AP}=\dfrac{m}{m+n}\overrightarrow{AB}\). Starting at \(A\),

\(\overrightarrow{OP}=\mathbf{a}+\dfrac{m}{m+n}(\mathbf{b}-\mathbf{a})\).

The fraction uses the part travelled from the starting point, not the remaining part. For a midpoint use \(\dfrac12\). A multiplier greater than 1 continues beyond \(B\); a negative multiplier travels in the opposite direction.

In a proof, derive a midpoint or ratio from the givens before using it. For area ratios, triangles with the same perpendicular height have areas in the ratio of their bases; no squared scale factor is needed.

Division ratios and proof

Division ratios and proof