Dividing a Segment with Vector Scaling

Introduction

A straight segment can be described by its endpoints and the displacement from one endpoint to the other. Scaling that displacement by a number between zero and one gives a precise way to locate every point on the segment, including points that divide it into equal parts or in a specified ratio.

This method works in three dimensions just as it does in two. It uses vector subtraction to find the displacement and scalar multiplication to take the required fraction of it. The key is to scale the displacement, not the endpoint coordinates independently.

From Endpoints to a Point on the Segment

Let \(A\) and \(B\) be points in three-dimensional space, with position vectors \(\mathbf{a}\) and \(\mathbf{b}\). The displacement from \(A\) to \(B\) is \(\mathbf{b}-\mathbf{a}\). Starting at \(A\), moving through a fraction \(t\) of this displacement gives a point \(P\) with position vector \(\mathbf{p}\).

The vector \(\mathbf{b}-\mathbf{a}\) gives the full displacement. Multiplying it by \(t\) gives the displacement from \(A\) to \(P\). Thus \(\mathbf{p}-\mathbf{a}=t(\mathbf{b}-\mathbf{a})\). The restriction \(0\le t\le1\) is what keeps \(P\) on the segment: values outside this interval would place the point on the line beyond one of the endpoints.

Equivalently, the formula can be written as a weighted sum of the endpoint position vectors:

To see the equivalence, expand the displacement formula: \(\mathbf{a}+t(\mathbf{b}-\mathbf{a})=\mathbf{a}+t\mathbf{b}-t\mathbf{a}=(1-t)\mathbf{a}+t\mathbf{b}\). The first form highlights the scaled displacement; the second highlights the two endpoint vectors and their weights.

Why the Fraction Formula Works

The formula describes both the location and the fraction of the segment used. The following derivation makes the connection explicit. Assume \(A\ne B\), so the full displacement is nonzero and the segment has positive length. Since \(0\le t\le1\), scaling the displacement by \(t\) gives a displacement of length \(t\) times the full length. The remaining displacement from \(P\) to \(B\) is the rest of the original displacement.

In more detail, the length statements follow from the magnitude rule \(\lVert c\mathbf{v}\rVert=|c|\lVert\mathbf{v}\rVert\). Here both scale factors \(t\) and \(1-t\) are nonnegative. Therefore, \(AP=t\,AB\) and \(PB=(1-t)\,AB\), where \(AP\), \(PB\), and \(AB\) denote the corresponding segment lengths. Adding gives \(AP+PB=AB\), as expected for a point on the segment.

When a point divides a nondegenerate segment internally in the ratio \(AP:PB=m:n\), where \(m,n>0\), its parameter is \(t=m/(m+n)\). Indeed, the fraction of the whole length used from \(A\) is \(m/(m+n)\), while the remaining fraction is \(n/(m+n)\). The corresponding vector formula is:

The weight on \(\mathbf{b}\) is \(m/(m+n)\), not \(n/(m+n)\): a point closer to \(A\) uses a smaller fraction of the displacement from \(A\) to \(B\).

Worked Example: Finding a Point One Third of the Way Along

Worked Example: Dividing a Segment in a Given Ratio

Worked Example: Dividing a Segment into Equal Parts

Component Form and a Common Mistake

If \(\mathbf{a}=(a_1,a_2,a_3)\) and \(\mathbf{b}=(b_1,b_2,b_3)\), the point at fraction \(t\) has coordinates

Each coordinate changes by the same fraction of its corresponding displacement component. This does not mean one should simply add a scaled endpoint to the starting point.

For a concrete check, take \(A=(2,1,0)\), \(B=(6,5,4)\), and \(t=1/2\). The incorrect calculation gives \(\mathbf{p}=(2,1,0)+\frac12(6,5,4)=(5,7/2,2)\). But its displacement from \(A\) is \((3,5/2,2)\), which is not a scalar multiple of the correct full displacement \((4,4,4)\). It is therefore not on the segment. Correctly, first subtract the endpoints: \(\mathbf{b}-\mathbf{a}=(4,4,4)\). Then \(\mathbf{p}=(2,1,0)+\frac12(4,4,4)=(4,3,2)\). The check gives \(\mathbf{p}-\mathbf{a}=(2,2,2)=\frac12(4,4,4)\) and \(\mathbf{b}-\mathbf{p}=(2,2,2)\), so this is the midpoint.

Conditions and Interpretation

The segment formula assumes that \(A\) and \(B\) are fixed endpoints and that \(t\) lies between zero and one. If \(A=B\), then \(\mathbf{b}-\mathbf{a}=\mathbf{0}\), so the formula gives \(\mathbf{p}=\mathbf{a}\) for every \(t\). There is no nonzero segment to divide, and a ratio of its two part lengths is undefined.

For distinct endpoints, the parameter gives a unique point on the segment: different values of \(t\) produce different points because their displacements from \(A\) are different scalar multiples of the same nonzero vector. This makes the parameter useful whenever a problem specifies a fraction, a ratio, or a number of equal subdivisions. It also provides a direct check: subtract the starting endpoint and compare the result with the specified fraction of the full displacement.