Why is the dot product of two different unit vectors 0?

The unit vectors e1, e2, e3 are perpendicular. By the definition e1·e2 = 1·1·cos 90° = 0. For equal unit vectors the angle is 0°, so e1·e1 = 1.

How do you derive the coordinate formula of the dot product?

Write a = a1e1 + a2e2 and b = b1e1 + b2e2, insert and multiply out. You get terms ai bj (ei·ej). Only those with i = j are 1, all others 0. What remains is a1b1 + a2b2.

Does the formula also hold in space?

Yes. With three unit vectors there are 9 terms, six of them are 0. What remains is a·b = a1b1 + a2b2 + a3b3.

What can I set in the app?

Plane or space, a numerical example at the end, play, and step forwards and backwards.

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Dot product: coordinate formula from the definition

About this app

Why a·b = a1b1 + a2b2 follows from a·b = |a||b|cos φ: decompose into unit vectors, insert, multiply out, evaluate the boxes e_i·e_j with cos 0° = 1 and cos 90° = 0 — the zero terms drop out. In 2D (4 terms) and 3D (9 terms), optionally with a numerical example.

With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.

Two formulas for the same dot product

The dot product is defined through lengths and the angle: a·b = |a|·|b|·cos φ. In practice it is computed with the coordinate formula a·b = a1b1 + a2b2 (+ a3b3 in space). Both give the same value for a reason: the coordinate formula follows from the definition and the rules swap, pull out numbers and multiply out. Conversely it gives the angle: cos φ = a·b / (|a|·|b|).

What the app shows

The derivation runs step by step as an animated formula. Both vectors are decomposed into unit vectors; a small sketch shows e1 and e2 with length 1 and a right angle. After inserting, everything is multiplied out term by term. Boxes frame the dot products of the unit vectors; with the definition they become 1 (equal vectors, cos 0°) or 0 (perpendicular, cos 90°). The zero terms drop out and the coordinate formula remains.

Try it

Use the arrows to go forwards and backwards; every transition also runs in reverse. In space there are 9 terms instead of 4, six of them drop out. A numerical example checks (1,2,3)·(4,5,6) = 4 + 10 + 18 = 32.

Frequently asked questions

Why is the dot product of two different unit vectors 0?: The unit vectors e1, e2, e3 are perpendicular. By the definition e1·e2 = 1·1·cos 90° = 0. For equal unit vectors the angle is 0°, so e1·e1 = 1.
How do you derive the coordinate formula of the dot product?: Write a = a1e1 + a2e2 and b = b1e1 + b2e2, insert and multiply out. You get terms ai bj (ei·ej). Only those with i = j are 1, all others 0. What remains is a1b1 + a2b2.
Does the formula also hold in space?: Yes. With three unit vectors there are 9 terms, six of them are 0. What remains is a·b = a1b1 + a2b2 + a3b3.
What can I set in the app?: Plane or space, a numerical example at the end, play, and step forwards and backwards.

Subject: Analytic geometry

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