Magnitude and Unit Vectors

To obtain a vector of prescribed length \(L>0\) in the direction of a nonzero vector \(\mathbf a\), first divide by \(|\mathbf a|\), then multiply by \(L\). The two vectors of length \(L\) parallel to \(\mathbf a\) are \(\pm \dfrac{L\mathbf a}{|\mathbf a|}\). In the same direction selects the positive sign.

For \(\mathbf a=(2,-1,2)\), \(|\mathbf a|=3\) and its unit direction is \(\left(\dfrac23,-\dfrac13,\dfrac23\right)\). Multiplying this by \(6\) gives \((4,-2,4)\), whose magnitude is \(6\). The zero vector cannot be normalized.

For points \(A\) and \(B\), form \(\overrightarrow{AB}=\mathbf b-\mathbf a\) before normalizing. A point a distance \(L\) from \(A\) on the line \(AB\) can have position vector \(\mathbf a\pm \dfrac{L(\mathbf b-\mathbf a)}{|\mathbf b-\mathbf a|}\); a direction condition chooses one.