Vector Addition and the Parallelogram Law

Part 10, Chapter 10: Vectors and Operations on Points

Learning objectives

Add and subtract points (vectors) in the coordinate plane
Understand the parallelogram law geometrically
Apply scalar multiplication to point operations

Points in the plane can be added. That sentence sounds strange the first time you read it, points are locations, not numbers. But once we have coordinates, adding two points coordinate-wise yields a third point with a clean geometric meaning: the diagonal of a parallelogram. This single observation upgrades the plane from a set of locations to an algebraic vector space, the same structure that powers physics, linear algebra, and computer graphics.

The algebra: addition and subtraction

For $P = (x_1, y_1)$ and $Q = (x_2, y_2)$ , define

$P + Q = (x_1 + x_2,\; y_1 + y_2),\qquad P - Q = (x_1 - x_2,\; y_1 - y_2).$

That is, add or subtract coordinate-wise. Combined with scalar multiplication $cP = (cx_1, cy_1)$ from the previous section, we have a full algebra. These satisfy familiar laws: addition is commutative ( $P + Q = Q + P$ ), associative ( $(P + Q) + R = P + (Q + R)$ ); the zero vector is $O = (0, 0)$ ; the additive inverse of $P = (x, y)$ is $-P = (-x, -y)$ ; and scalar multiplication distributes: $c(P + Q) = cP + cQ$ and $(c + d)P = cP + dP$ .

The geometry: parallelogram law

Now the geometric meaning. Given two points $P$ and $Q$ (and the origin $O$ ), the four points

$O,\; P,\; Q,\; P + Q$

form the vertices of a parallelogram. The point $P + Q$ sits at the vertex diagonally opposite to $O$ . Equivalently, to compute $P + Q$ geometrically: translate $OQ$ to start at $P$ ; its endpoint lands on $P + Q$ . This is the parallelogram law, the geometric face of vector addition.

Where this shows up

Physics: Forces add as vectors: two ropes pulling on a heavy box with forces $\vec{F_1}$ and $\vec{F_2}$ produce a net force $\vec{F_1} + \vec{F_2}$ , found by drawing the parallelogram of forces, that diagram is exactly the parallelogram law.

Sailing: A sailboat's velocity-made-good against the wind is computed by parallelogram-adding the boat's velocity through the water and the current's velocity over ground.

Computer Animation: Blending two animations (e.g., "walking" and "reaching") uses linear combinations of pose vectors: $\alpha \cdot \vec{p}_{walk} + (1-\alpha) \cdot \vec{p}_{reach}$ is parallelogram-law interpolation in joint-angle space.

Drag $P$ and $Q$ . The widget draws the four-vertex parallelogram and labels $P + Q$ . The accent diagonal from the origin to $P + Q$ is the sum vector.

Subtraction = displacement

The difference $P - Q$ represents the displacement vector from $Q$ to $P$ . To move from $Q$ to $P$ , you must translate by $P - Q$ . Its length is the distance $|P - Q|$ between the two points, the same distance formula from §8.2. So distance, midpoint, and vector subtraction are three views of one operation.

Linear combinations

Combining scalar multiplication with addition produces all linear combinations $aP + bQ$ . These are the building blocks of every line and plane in the textbook. A line through the origin in direction $P$ consists of all $tP$ for $t \in \mathbb{R}$ ; an arbitrary line through $P$ in direction $Q$ consists of all $P + tQ$ . Everything geometric we will do for the rest of the book is built on these two operations.

Try it

Predict first: $P + Q$ for $P = (3, -2)$ and $Q = (-1, 5)$ is what? Place both vectors in the widget and verify the sum lights up at $(2, 3)$ , the parallelogram's far corner.

If three vertices of a parallelogram are $O$ , $A = (4, 1)$ , $B = (2, 3)$ , the fourth vertex is $A + B = (6, 4)$ .

Predict first: what should $P + (-P)$ equal for any $P$ ? Place $P = (7, -3)$ and verify $P + (-P)$ lands at the origin, the additive inverse rule for vectors.

A trap to watch for

The parallelogram law gives $P + Q$ as the opposite vertex from $O$ , not as one of the adjacent vertices. A common error is to draw $P$ and $Q$ and then label one of them as $P + Q$ . The sum lives at the corner that the parallelogram completes, diagonally across from the origin. Sanity check: $P + Q$ should be the vertex whose position vector from $O$ equals the sum of the position vectors of $P$ and $Q$ . If your "sum" coincides with $P$ or $Q$ or $O$ , something is wrong.

What you now know

You can add, subtract, and scale points, and you can interpret each operation geometrically, addition as the parallelogram law, subtraction as the displacement between two points. These laws turn the plane into a vector space over $\mathbb{R}$ , the foundation for the next chapter on segments, rays, and lines (and ultimately for all of linear algebra).

Quick check

Mark section complete →

References

Lang, S. (1971). Basic Mathematics. Springer. Chapter 9, §2, point addition and the parallelogram law.

Halmos, P. R. (1958). Finite-Dimensional Vector Spaces, 2nd ed. Springer. §1 abstracts the parallelogram-law structure into the axioms of a vector space.

Coxeter, H. S. M. (1969). Introduction to Geometry, 2nd ed. Wiley. §13 covers affine transformations, the broader framework.

Artin, M. (2011). Algebra, 2nd ed. Pearson. Chapter 3 develops vector spaces in the context of linear algebra.

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