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Why Vector Addition Follows the Parallelogram Rule

See why adding two vectors from a common starting point produces the diagonal of a parallelogram, and practise finding that diagonal from coordinates.

Beginner 10 min read

What You'll Learn

  • Explain how component addition determines the endpoint of a resultant vector.
  • Construct a parallelogram from two vectors with a common starting point.
  • Find the fourth vertex and diagonal using three-dimensional coordinates.
  • Check that opposite sides have matching directed displacements.
  • Recognize when the parallelogram construction collapses to a line.
  • Avoid confusing endpoint coordinates with displacement vectors.

From Component Addition to a Geometric Rule

When two vectors begin at the same point, their sum has a simple geometric interpretation: it is the diagonal that starts at that point and reaches the opposite corner of a parallelogram. This rule is useful because it connects a coordinate calculation with a picture of how two directed displacements combine.

The picture is not an extra rule that competes with component addition. It follows from component addition. We will use coordinates to find the fourth vertex, check the directed displacements along the sides, and identify which diagonal represents the sum.

The Setup

Suppose two vectors \(\mathbf{u}\) and \(\mathbf{v}\) start at the same point \(P\). Let their endpoints be \(A\) and \(B\), respectively. In other words, moving from \(P\) to \(A\) gives displacement \(\mathbf{u}\), and moving from \(P\) to \(B\) gives displacement \(\mathbf{v}\).

Write the points and vectors in coordinates as

$$ P=(p_1,p_2,p_3),\qquad \mathbf{u}=(u_1,u_2,u_3),\qquad \mathbf{v}=(v_1,v_2,v_3). $$

Adding a displacement to a point means adding its components to the point's coordinates. Thus the endpoints are \(A=P+\mathbf{u}\) and \(B=P+\mathbf{v}\). Component addition gives the sum vector

$$ \mathbf{u}+\mathbf{v} = (u_1+v_1,\;u_2+v_2,\;u_3+v_3). $$

Starting at \(P\) and following this sum gives a point \(C\). Its coordinates are the coordinates of \(P\) plus the components of the sum.

$$ C=P+(\mathbf{u}+\mathbf{v}). $$
Definition: The parallelogram rule for adding two vectors with a common starting point is to construct the parallelogram whose two sides from that point represent the vectors. Their sum is represented by the diagonal from the common starting point to the opposite vertex.

Why the Diagonal Is the Sum

Here is the coordinate argument behind the rule. The fourth vertex must be the endpoint reached by starting at \(P\) and applying both displacements. By component addition, that endpoint is \(C=P+(\mathbf{u}+\mathbf{v})\). The crucial check is that the other two sides of the quadrilateral have the same directed displacements as the original sides.

$$ \begin{aligned} C-A &=[P+(\mathbf{u}+\mathbf{v})]-(P+\mathbf{u}) =\mathbf{v},\\ C-B &=[P+(\mathbf{u}+\mathbf{v})]-(P+\mathbf{v}) =\mathbf{u}. \end{aligned} $$

So the side from \(A\) to \(C\) has displacement \(\mathbf{v}\), matching the side from \(P\) to \(B\). The side from \(B\) to \(C\) has displacement \(\mathbf{u}\), matching the side from \(P\) to \(A\). These pairs of opposite sides are translated copies of one another, so \(P,A,C,B\), in that order, form a parallelogram. The diagonal from \(P\) to \(C\) has displacement \(\mathbf{u}+\mathbf{v}\), as claimed.

Theorem: If \(\mathbf{u}\) and \(\mathbf{v}\) start at the same point \(P\), the opposite vertex of the parallelogram is \(C=P+(\mathbf{u}+\mathbf{v})\). The diagonal from \(P\) to \(C\) represents the sum \(\mathbf{u}+\mathbf{v}\). If both vectors lie along the same line, the construction collapses to a line, but the coordinate formula for the sum still holds.

This derivation explains why the rule works: component addition gives the proposed opposite vertex, and subtracting endpoint coordinates confirms that the other two sides repeat the original directed displacements. The rule does not depend on how the diagram is drawn or on whether the vectors point in a particular direction.

Worked Example: Constructing the Parallelogram in Three Dimensions

Worked Example: Constructing the Parallelogram in Three Dimensions

Let the common starting point be \(P=(1,2,0)\), and let the two vectors be \(\mathbf{u}=(2,1,1)\) and \(\mathbf{v}=(-1,3,2)\). Find their endpoints, the opposite vertex, and the diagonal displacement.

First find the endpoint \(A\) of \(\mathbf{u}\) by adding its components to \(P\):

$$ A=P+\mathbf{u} =(1+2,\;2+1,\;0+1) =(3,3,1). $$

Next find the endpoint \(B\) of \(\mathbf{v}\):

$$ B=P+\mathbf{v} =(1-1,\;2+3,\;0+2) =(0,5,2). $$

Add the vectors component by component to obtain the diagonal displacement:

$$ \mathbf{u}+\mathbf{v} =(2-1,\;1+3,\;1+2) =(1,4,3). $$

Starting this displacement at \(P\) gives the opposite vertex:

$$ C=P+(\mathbf{u}+\mathbf{v}) =(1+1,\;2+4,\;0+3) =(2,6,3). $$

Check both other sides. From \(A\) to \(C\), the displacement is

$$ C-A=(2-3,\;6-3,\;3-1)=(-1,3,2)=\mathbf{v}. $$

From \(B\) to \(C\), it is

$$ C-B=(2-0,\;6-5,\;3-2)=(2,1,1)=\mathbf{u}. $$

Both checks match the original directed side displacements. Thus \(P,A,C,B\) form the parallelogram, and \(P\)-to-\(C\) is its diagonal representing the sum \((1,4,3)\).

Worked Example: When Components Cancel

Worked Example: When Components Cancel

Take \(P=(0,0,0)\), \(\mathbf{u}=(3,-2,1)\), and \(\mathbf{v}=(1,2,0)\). Find the fourth vertex and explain what the sum tells us.

The two side endpoints are

$$ A=P+\mathbf{u}=(0+3,\;0-2,\;0+1)=(3,-2,1), $$
$$ B=P+\mathbf{v}=(0+1,\;0+2,\;0+0)=(1,2,0). $$

Adding the vectors gives

$$ \mathbf{u}+\mathbf{v} =(3+1,\;-2+2,\;1+0) =(4,0,1). $$

Therefore the opposite vertex is

$$ C=P+(\mathbf{u}+\mathbf{v})=(0+4,\;0+0,\;0+1)=(4,0,1). $$

The second components cancel, so the resultant has no change in the second coordinate, even though each original vector has a nonzero second component. Check the construction by subtracting endpoint coordinates:

$$ C-A=(4-3,\;0-(-2),\;1-1)=(1,2,0)=\mathbf{v}, $$
$$ C-B=(4-1,\;0-2,\;1-0)=(3,-2,1)=\mathbf{u}. $$

The opposite sides have the required displacements, and the diagonal from \(P\) to \(C\) is exactly the component sum \((4,0,1)\).

Worked Example: Starting Away from the Origin

Worked Example: Starting Away from the Origin

Suppose \(P=(2,0,-1)\), \(\mathbf{u}=(2,-1,0)\), and \(\mathbf{v}=(-1,2,1)\). Find the opposite vertex and verify the parallelogram rule.

The two endpoints from the common starting point are

$$ A=P+\mathbf{u}=(2+2,\;0-1,\;-1+0)=(4,-1,-1), $$
$$ B=P+\mathbf{v}=(2-1,\;0+2,\;-1+1)=(1,2,0). $$

The sum vector and its endpoint are

$$ \mathbf{u}+\mathbf{v} =(2-1,\;-1+2,\;0+1)=(1,1,1), $$
$$ C=P+(\mathbf{u}+\mathbf{v}) =(2+1,\;0+1,\;-1+1)=(3,1,0). $$

Now check the side displacements:

$$ C-A=(3-4,\;1-(-1),\;0-(-1))=(-1,2,1)=\mathbf{v}, $$
$$ C-B=(3-1,\;1-2,\;0-0)=(2,-1,0)=\mathbf{u}. $$

The checks confirm that the two pairs of opposite sides carry the original directed displacements. The diagonal from \(P=(2,0,-1)\) to \(C=(3,1,0)\) represents \((1,1,1)\).

A Common Mistake: Adding Endpoint Coordinates

The vectors are the displacements from \(P\) to the endpoints, not the endpoint coordinates themselves. For example, in the first worked example, a mistaken calculation might add \(A\) and \(B\) directly:

$$ (3,3,1)+(0,5,2)=(3,8,3). $$

This does not give the opposite vertex. It adds two point coordinates, rather than adding the two displacements from their common starting point. The correct calculation is to identify the vectors as \(A-P=(2,1,1)\) and \(B-P=(-1,3,2)\), add those displacements, and then start their sum at \(P\):

$$ P+[(A-P)+(B-P)] =(1,2,0)+(1,4,3) =(2,6,3). $$

A reliable check is to subtract the coordinates of each endpoint from the proposed opposite vertex. The resulting displacements must reproduce the other two sides.

Interpreting the Rule

The parallelogram picture shows how two displacements combine at a glance. Each vector is one side beginning at \(P\); the sum is the diagonal beginning at \(P\). This makes it possible to interpret a component calculation geometrically without changing the underlying arithmetic.

To apply the rule in coordinates, keep three objects distinct: the common starting point, the two displacement vectors, and the endpoints reached by those displacements. Add the vector components to find the diagonal displacement, then add that displacement to the starting point to locate the opposite vertex. If the two vectors run along the same line, the parallelogram flattens, but the component calculation and diagonal interpretation remain valid.

Check Your Understanding

Use coordinates to answer these questions. In each case, verify the result by checking the directed displacements of the opposite sides.

  1. Starting at \(P=(1,0,2)\), two vectors are \(\mathbf{u}=(2,1,-1)\) and \(\mathbf{v}=(0,3,2)\). What is their sum and the opposite vertex?
  2. Why does the displacement from the endpoint of \(\mathbf{u}\) to the opposite vertex equal \(\mathbf{v}\)?
  3. If the second components of two vectors are \(5\) and \(-5\), what is the second component of their sum?
  4. What is the error in adding the coordinates of two endpoints to find the parallelogram's fourth vertex?
  5. What happens to the geometric construction if the two vectors lie along the same line?

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