Recovering a Missing Leg in a Displacement Chain

Introduction

A route can consist of several directed legs, while the displacement from its starting point to its endpoint is known. If all but one leg are given, the missing leg is determined by the total: subtract the known legs from that total.

This is not a calculation with distances alone. Each leg has both a size and a direction, so it must be represented by a vector. A leg that points in the negative direction of an axis has a negative component, and that sign must be preserved during the subtraction.

The method is useful whenever a route description is incomplete but its overall change in position is known. The key is to write the complete vector equation first, then isolate the unknown vector.

Displacement Chains as Vector Equations

A displacement vector describes the change in position from the beginning of a leg to its end. In three dimensions, it can be written using its components, such as \(\mathbf{a}=\langle a_x,a_y,a_z\rangle\). The components are directed quantities: a negative component means motion in the negative direction of the corresponding coordinate axis.

If a route has successive legs \(\mathbf{a},\mathbf{u},\mathbf{c}\), then its total displacement is their vector sum. The order follows the route, but vector addition is commutative, so the sum can be rearranged when isolating a leg.

The formula applies when the given total and legs describe the same route, in the same coordinate system, and use consistent units. It finds the vector for the missing leg, including its direction. It does not merely give the distance traveled along that leg.

Vector subtraction is component-wise. If \(\mathbf{p}=\langle p_x,p_y,p_z\rangle\) and \(\mathbf{q}=\langle q_x,q_y,q_z\rangle\), then \(\mathbf{p}-\mathbf{q}=\langle p_x-q_x,p_y-q_y,p_z-q_z\rangle\). In a displacement chain, first add all known legs component by component; then subtract that sum from the total.

Worked Example: One Missing Leg in a Three-Leg Route

Why Subtraction Recovers the Missing Leg

The subtraction rule follows directly from the vector equation for the route. Let \(\mathbf{K}\) be the sum of all the known legs. Since the total consists of those known legs and the missing leg, \(\mathbf{D}=\mathbf{K}+\mathbf{u}\). Add the opposite vector \(-\mathbf{K}\) to both sides. Vector addition has an additive identity \(\mathbf{0}\), and \(-\mathbf{K}+\mathbf{K}=\mathbf{0}\). Therefore the known contribution cancels on the right:

Thus \(\mathbf{u}=\mathbf{D}-\mathbf{K}\). This reasoning uses the additive inverse of a vector, not an operation on its length. In components, the same cancellation occurs separately in each coordinate. That is why the formula works no matter how many known legs there are.

Worked Example: Keeping Negative Components Straight

A Route with Several Known Legs

The same method works if the missing leg is surrounded by several known legs, or if it appears first or last in the route. The route equation includes every leg. Then the sum of all known vectors is subtracted from the total. The calculation does not depend on the unknown leg’s position in the written sequence, because addition of vectors is commutative and associative.

What the Answer Tells You

A recovered vector describes the change in position required to make the route equation true. For example, a zero component means there is no net change along that coordinate direction during that leg. A negative component indicates a change in the negative coordinate direction. Neither sign tells you that the leg’s length is negative; lengths are nonnegative, while vector components are signed.

The total displacement does not record every detail of the route. Different routes can have the same total displacement. Here, the missing leg is determined because every other leg is specified. If two or more leg vectors are unknown, the total generally determines only their combined sum, not each vector separately. Additional information would be needed to determine them individually.

Units should also agree throughout the calculation. Do not add a displacement in metres to one in kilometres without first converting them to the same unit. The component arithmetic can be correct and still produce an invalid result if the quantities being combined use incompatible units.

Common Mistake: Subtracting in the Wrong Direction

A frequent error is to subtract the total displacement from the sum of the known legs. That reverses the vector subtraction and gives the negative of the required answer.

For instance, in the four-leg example, the known-leg sum is \(\langle 5,2,0\rangle\) m and the total is \(\langle 8,2,-6\rangle\) m. The incorrect calculation would be \(\langle 5,2,0\rangle-\langle 8,2,-6\rangle=\langle -3,0,6\rangle\) m.

But adding that incorrect vector to the known legs gives \(\langle 5,2,0\rangle+\langle -3,0,6\rangle=\langle 2,2,6\rangle\) m, not the required total \(\langle 8,2,-6\rangle\) m. The correct order is total minus known-leg sum: \(\langle 8,2,-6\rangle-\langle 5,2,0\rangle=\langle 3,0,-6\rangle\) m. Adding this result to the known legs does give \(\langle 8,2,-6\rangle\) m.