Vector Operations

A delivery drone lifts off from a warehouse in Bengaluru and flies 3 km due east. Then it turns and flies 4 km due north. How far is it from the warehouse now — and in what direction?

You could solve this with the Pythagorean theorem: \sqrt{3^2 + 4^2} = 5 km, at some angle north-east. That works. But the real power of the question is this: the drone's two separate legs of travel combine into a single net displacement. Three kilometres east and four kilometres north add up to five kilometres north-east. That "adding up" is not ordinary addition — you cannot just write 3 + 4 = 5. The directions matter. What you are doing is adding vectors.

Vector addition is the first operation you learn on vectors, and it is unlike anything you have done with plain numbers. When you add the number 3 to the number 4, you always get 7. When you add a vector of magnitude 3 to a vector of magnitude 4, you might get 7 (if they point in the same direction), or 1 (if they point in opposite directions), or 5 (if they are perpendicular), or anything in between. The result depends on the angle.

This article gives you the two fundamental laws for adding vectors, proves their key properties, shows you how scalar multiplication works, and then uses both operations to derive the section formula — one of the most useful results in coordinate geometry.

Adding vectors: the triangle law

Here is the rule, stated as a picture first.

To add vector \vec{a} to vector \vec{b}: place the tail of \vec{b} at the head of \vec{a}. The vector from the tail of \vec{a} to the head of \vec{b} is the sum \vec{a} + \vec{b}.

The name "triangle law" comes from the shape the three vectors make. The two summands form two sides of a triangle, and the sum is the third side.

Notice something important: the sum \vec{a} + \vec{b} skips the intermediate stop. The drone flew east, then north — but the net effect is the single diagonal vector from start to finish. Vector addition compresses a multi-step journey into one step.

The parallelogram law

There is a second way to add the same two vectors, and it gives the same answer.

Place \vec{a} and \vec{b} so they share the same starting point. Complete the parallelogram — draw the copy of \vec{a} starting from the head of \vec{b}, and the copy of \vec{b} starting from the head of \vec{a}. The diagonal of the parallelogram, starting from the common point, is \vec{a} + \vec{b}.

Why does this give the same answer as the triangle law? Look at the upper triangle in the parallelogram: it has \vec{a} along the bottom, the copy of \vec{b} along the right side, and the diagonal as the third side. That is the triangle law applied to \vec{a} and \vec{b}. The parallelogram law is just the triangle law drawn from a different starting configuration.

The parallelogram law is especially useful in physics, where two forces act on the same object from the same point and you want the net force. It is also the law that makes the commutativity of vector addition visually obvious — the other diagonal of the same parallelogram is \vec{b} + \vec{a}, and both diagonals start from the same corner, so \vec{a} + \vec{b} = \vec{b} + \vec{a}.

Properties of vector addition

Vector addition satisfies the same core properties as ordinary addition. Each one is worth proving, because the proofs are short and they build geometric intuition.

Commutativity: \vec{a} + \vec{b} = \vec{b} + \vec{a}

Proof. Construct the parallelogram with \vec{a} and \vec{b} starting from the same point O. The diagonal from O to the opposite corner is \vec{a} + \vec{b} (reading through the triangle O \to A \to P) and also \vec{b} + \vec{a} (reading through the triangle O \to B \to P). Both paths end at the same point P, so the resultant vector is the same.

In other words, commutativity is a consequence of the fact that a parallelogram has a unique diagonal from each pair of opposite corners. The order in which you traverse the two sides does not change where you end up.

Associativity: (\vec{a} + \vec{b}) + \vec{c} = \vec{a} + (\vec{b} + \vec{c})

Proof. Place the three vectors head-to-tail: \vec{a} from O to A, \vec{b} from A to B, \vec{c} from B to C.

The sum (\vec{a} + \vec{b}) + \vec{c}: first add \vec{a} and \vec{b} to get the vector \overrightarrow{OB}, then add \vec{c} to get \overrightarrow{OC}.

The sum \vec{a} + (\vec{b} + \vec{c}): first add \vec{b} and \vec{c} to get the vector \overrightarrow{AC}, then add \vec{a} to get \overrightarrow{OC}.

Both groupings produce \overrightarrow{OC}. The endpoint does not depend on how you bracket the sum — only on the final chain from O to C. \square

This extends to any number of vectors: \vec{a}_1 + \vec{a}_2 + \cdots + \vec{a}_n is unambiguous regardless of how you group them. Place them all head-to-tail; the sum is the vector from the first tail to the last head.

Additive identity: \vec{a} + \vec{0} = \vec{a}

The zero vector \vec{0} has magnitude zero and no specific direction. Adding it to any vector \vec{a} using the triangle law means placing \vec{0} at the head of \vec{a} — but \vec{0} goes nowhere, so the sum ends at the same point as \vec{a} itself. The result is \vec{a}, unchanged.

Additive inverse: \vec{a} + (-\vec{a}) = \vec{0}

The vector -\vec{a} has the same magnitude as \vec{a} but points in the opposite direction. Place -\vec{a} at the head of \vec{a} using the triangle law. You walk forward along \vec{a}, then walk the same distance backward along -\vec{a}, ending up exactly where you started. The net displacement is \vec{0}.

This also defines subtraction: \vec{a} - \vec{b} means \vec{a} + (-\vec{b}). Geometrically, reverse \vec{b} and then add.

Scalar multiplication

Multiplying a vector by a number (a scalar) is simpler than vector addition. It changes the vector's length without altering its line of action.

Think of scalar multiplication as a zoom knob. Turning it up (k > 1) stretches the vector. Turning it down (0 < k < 1) shrinks it. Flipping it negative (k < 0) reverses and then stretches.

Properties of scalar multiplication

These follow directly from the definition and are essential for algebraic manipulation of vectors.

1. Distributive over vector addition: k(\vec{a} + \vec{b}) = k\vec{a} + k\vec{b}.

Proof. Consider the triangle formed by \vec{a}, \vec{b}, and \vec{a} + \vec{b}. Scale the entire triangle by factor k. Each side scales by k, so \vec{a} becomes k\vec{a}, \vec{b} becomes k\vec{b}, and \vec{a} + \vec{b} becomes k(\vec{a} + \vec{b}). The scaled triangle still satisfies the triangle law, so k\vec{a} + k\vec{b} = k(\vec{a} + \vec{b}).

2. Distributive over scalar addition: (k + l)\vec{a} = k\vec{a} + l\vec{a}.

Proof. The vector (k + l)\vec{a} has magnitude |k + l| \cdot |\vec{a}| along the line of \vec{a}. The vectors k\vec{a} and l\vec{a} both lie along the same line (or its opposite). Placed head-to-tail, their magnitudes add as real numbers: k|\vec{a}| + l|\vec{a}| = (k+l)|\vec{a}|. The direction is determined by the sign of k + l, which matches.

3. Associative with scalars: k(l\vec{a}) = (kl)\vec{a}.

Proof. The vector l\vec{a} has magnitude |l||\vec{a}|. Scaling again by k gives magnitude |k| \cdot |l| \cdot |\vec{a}| = |kl| \cdot |\vec{a}|. The direction: if both k and l are positive, the direction stays. If one is negative, the direction flips once. If both are negative, the direction flips twice and returns to the original. In every case, the direction matches that of (kl)\vec{a}.

4. Identity: 1 \cdot \vec{a} = \vec{a}.

This is immediate from the definition: magnitude |1| \cdot |\vec{a}| = |\vec{a}|, direction unchanged.

Subtraction, visually

Subtraction deserves its own picture. To compute \vec{a} - \vec{b}, you reverse \vec{b} and add.

In the parallelogram picture, if \vec{a} and \vec{b} both start from the same point, then \vec{a} + \vec{b} is the diagonal from that common point, and \vec{a} - \vec{b} is the other diagonal — the one running from the head of \vec{b} to the head of \vec{a}.

This picture is worth remembering. Every time you see a parallelogram, its two diagonals encode a sum and a difference.

The section formula

You now have two tools: addition and scalar multiplication. Here is a beautiful application that combines them.

The problem. Given two points A and B with position vectors \vec{a} and \vec{b} (relative to some origin O), find the position vector of the point P that divides the segment AB in the ratio m : n.

"Divides in the ratio m : n" means AP : PB = m : n. The point P sits on the segment AB, closer to A if m < n, closer to B if m > n, and at the midpoint if m = n.

Derivation

Let the position vector of P be \vec{p}. Then:

Since P divides AB in the ratio m : n, the vectors \overrightarrow{AP} and \overrightarrow{PB} point in the same direction (both from A toward B) and their magnitudes satisfy |\overrightarrow{AP}| / |\overrightarrow{PB}| = m/n. Because they are parallel and point the same way, the vector equation is:

which means:

Substitute the expressions in terms of position vectors:

Expand both sides:

Collect \vec{p} terms on one side:

Read this formula carefully. The coefficient of \vec{b} is m — the ratio component closest to A. The coefficient of \vec{a} is n — the ratio component closest to B. This cross-pattern is easy to remember: the "weight" on each endpoint is the ratio part belonging to the other end.

Special case: the midpoint. When m = n, the point P is exactly halfway between A and B:

This is the vector version of the midpoint formula from coordinate geometry.

External division

If P divides AB externally in the ratio m : n — meaning P lies on the line through A and B but outside the segment — the derivation is nearly identical, except \overrightarrow{AP} and \overrightarrow{PB} point in opposite directions. The sign change gives:

The minus signs appear because external division means walking past one of the endpoints.

Worked examples

Common confusions

• "Vector addition is the same as adding magnitudes." Only when both vectors point in exactly the same direction. In general, |\vec{a} + \vec{b}| \leq |\vec{a}| + |\vec{b}|, with equality only in the parallel case. This is the triangle inequality — the third side of a triangle is always shorter than the sum of the other two.

• "The parallelogram law and the triangle law give different answers." They give exactly the same answer — they are two ways of drawing the same addition. The parallelogram law starts both vectors at the same point; the triangle law chains them head-to-tail. The resultant is the same vector either way.

• "In the section formula, the coefficient of \vec{a} is m." It is not. The coefficient of \vec{a} is n, and the coefficient of \vec{b} is m. The cross-pattern: m weights the far end, n weights the near end. Mixing this up is the most common error in section formula problems.

• "Scalar multiplication can change a vector's direction to any angle." Scalar multiplication can only keep the direction or reverse it. It cannot rotate a vector. To change the angle, you need vector addition or a more advanced operation like rotation matrices.

• "The zero vector has no direction, so it isn't a real vector." The zero vector is a valid vector — it is the additive identity. It has magnitude zero and its direction is undefined (or, equivalently, it is considered parallel to every direction). Without it, the algebraic structure of vectors would break: you could not define subtraction as \vec{a} - \vec{a}.

Going deeper

The polygon law of addition

The triangle law generalises immediately. If you have n vectors \vec{a}_1, \vec{a}_2, \ldots, \vec{a}_n, place them head-to-tail in sequence. The sum \vec{a}_1 + \vec{a}_2 + \cdots + \vec{a}_n is the single vector from the tail of \vec{a}_1 to the head of \vec{a}_n. The n vectors and their sum form an (n+1)-sided polygon (which may not close — it closes only if the sum is \vec{0}).

A closed polygon of vectors means the sum is zero. This is the equilibrium condition in physics: if n forces act on a particle and their vector sum is zero, the particle is in equilibrium. The polygon closes.

The centroid via the section formula

The centroid of a triangle with vertices A(\vec{a}), B(\vec{b}), C(\vec{c}) is the point where the three medians intersect. You can find it by applying the section formula twice.

First, find the midpoint M of BC:

The median from A goes to M. The centroid G divides AM in the ratio 2:1 (a standard result from geometry). By the section formula:

The position vector of the centroid is the average of the three vertex position vectors. This is one of those results that is obviously "right" once you see it — the centroid is the democratic average of all three vertices, with each vertex contributing equally. The section formula made the derivation two lines long.

Vector spaces, briefly

The properties you proved earlier — commutativity, associativity, distributivity, existence of identity and inverse — are not just a list. They are exactly the axioms of a vector space. Any collection of objects that satisfies these rules (with an appropriate notion of scalar multiplication) is a vector space, even if the objects are not arrows in space.

Functions can be vectors: f + g is the function whose value at each point is f(x) + g(x). Polynomials can be vectors. Sequences can be vectors. The arrow-in-space picture is the first vector space you meet, but it is far from the last. The operations you learned in this article — addition and scalar multiplication — are the two primitive operations of all of linear algebra.

Where this leads next

• Components and Direction — how to break a vector into pieces along the coordinate axes, which turns geometric operations into arithmetic.
• Collinearity and Coplanarity — using the operations from this article to test whether points lie on the same line or the same plane.
• Dot Product — a new operation that multiplies two vectors and produces a number, not a vector. It measures the angle between them.
• Cross Product — another multiplication, this one producing a vector perpendicular to both inputs. It measures area.
• Coordinate Geometry Basics — the coordinate-geometry version of the section formula, with numerical coordinates instead of position vectors.