Vector Magnitudes as Segment Lengths

From a Vector to a Segment Length

A vector between two points records the change in position from one endpoint to the other. Its components may be positive, negative, or zero, depending on the direction of that change. The length of the segment joining the endpoints, however, cannot be negative.

The vector’s magnitude connects these two ideas: it gives the length of the corresponding directed segment. This interpretation lets us read a geometric distance directly from the components of a vector, while keeping clear which quantities describe position and which describe change in position.

Endpoints, Displacement, and Length

Let \(P=(x_1,y_1,z_1)\) and \(Q=(x_2,y_2,z_2)\) be points in three-dimensional Cartesian coordinates. The vector from \(P\) to \(Q\), written \(\mathbf{v}_{PQ}\), has components equal to the change in each coordinate:

The directed segment runs from \(P\) toward \(Q\). Its direction matters when describing the vector: reversing the endpoints reverses the direction. But the segment’s length, the distance from \(P\) to \(Q\), does not depend on which endpoint is named first.

The three-dimensional magnitude formula established earlier says that a vector \(\langle a,b,c\rangle\) has magnitude \(\sqrt{a^2+b^2+c^2}\). Applying it to the endpoint differences gives a practical distance formula:

Each coordinate difference is a signed change, not a segment length by itself. Squaring the differences ensures that changes in either direction contribute positively to the total length.

Why Magnitude Equals Segment Length

Here is the geometric reason for the connection. Write the coordinate changes as \(\Delta x=x_2-x_1\), \(\Delta y=y_2-y_1\), and \(\Delta z=z_2-z_1\). The coordinate axes give perpendicular directions, so the three component lengths are \(|\Delta x|\), \(|\Delta y|\), and \(|\Delta z|\).

First consider the right triangle formed by the horizontal coordinate changes in the \(xy\)-plane. If its diagonal has length \(r\), the Pythagorean theorem gives \(r^2=(\Delta x)^2+(\Delta y)^2\). The segment from \(P\) to \(Q\) is the diagonal of a right triangle whose other perpendicular side has length \(|\Delta z|\). Applying the theorem again gives \(d(P,Q)^2=r^2+(\Delta z)^2\). Substituting for \(r^2\) yields:

Distance is nonnegative, so taking the nonnegative square root gives \(d(P,Q)=\sqrt{(\Delta x)^2+(\Delta y)^2+(\Delta z)^2}\). This is exactly the magnitude of \(\mathbf{v}_{PQ}=\langle\Delta x,\Delta y,\Delta z\rangle\). The equality holds because both quantities measure the same spatial diagonal using the same perpendicular coordinate changes.

Worked Coordinate Examples

What Changes When the Direction Is Reversed?

If the endpoints are switched, each component of the vector changes sign. For example, \(\mathbf{v}_{QP}=\langle x_1-x_2,y_1-y_2,z_1-z_2\rangle=-\mathbf{v}_{PQ}\). But the squared components are unchanged, since \((-a)^2=a^2\). Consequently, the magnitude—and the segment length—is the same in either direction:

This does not mean the two vectors are equal: they point in opposite directions. It means that reversing the direction of traversal does not alter how long the segment is.

Common Mistake: Using an Endpoint Instead of a Difference

A frequent error is to calculate the magnitude of the endpoint \(Q\) and treat it as the length of the segment from \(P\) to \(Q\). The coordinates of \(Q\) describe its position relative to the origin. The segment from \(P\) to \(Q\), however, is described by the changes from \(P\) to \(Q\).

For example, let \(P=(2,-1,4)\) and \(Q=(5,3,4)\), with coordinates in meters. The incorrect calculation uses \(Q\) directly:

This is the distance from the origin to \(Q\), not the length of the segment \(PQ\). The correction is to find the displacement first:

Indeed, adding \(\langle 3,4,0\rangle\) to \(P\) gives \(Q\), and \(3^2+4^2=25\), so the length is \(5\,\text{m}\). The key question is always: Which two endpoints define the segment? Subtract the starting point from the ending point before finding the magnitude.

Why This Connection Matters

Components tell us how far a displacement goes along each coordinate axis, while magnitude combines those perpendicular changes into one overall length. This is useful whenever a problem gives endpoints but asks for a distance, or gives a vector and asks how long its corresponding segment is.

Keep the roles distinct: an endpoint is a location, a vector between endpoints is a change in location, and the vector’s magnitude is the nonnegative length of the segment traced by that change. Once the displacement is correct, the magnitude formula turns its components into a geometric length.