Right-handed systems in 2D and 3D
About this app
What makes (a, b) or (a, b, c) right-handed? A 3D hand fits itself to the vectors: thumb a, index finger b, middle finger c; in the plane the fingers curl from a to b. det > 0 as the clean definition, signed parallelepiped and area. Rotating keeps the orientation, swapping and reflecting reverse it, with a quiz.
With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.
What is a right-handed system?
An ordered triple (a, b, c) of linearly independent vectors is right-handed if det(a, b, c) = a · (b × c) > 0, otherwise left-handed. Intuitively, the right-hand rule applies: thumb along a, index finger along b, and the middle finger points to the side where c lies. In the plane, (a, b) is right-handed if det(a, b) = a₁b₂ − a₂b₁ > 0: a turns the shorter way counterclockwise to b. The orientation depends only on the order of the vectors.
What the app shows
A stylised hand fits itself to the vectors. In space, thumb, index and middle finger take on the colours of a, b and c; if the right hand does not fit, it shakes, and a faint arrow shows where c would have to be. In the plane, the fingers curl from a to b and the thumb points up or down. The parallelepiped and the parallelogram carry the colour of the sign, and the Definition panel computes the determinant live.
Try it yourself
Press "Rotate": the whole system turns and the determinant stays the same. Press "Swap" or "Reflect": on the way the vectors briefly lie flat in a plane, det becomes 0 and then changes sign. A cyclic shift (a, b, c) → (b, c, a) keeps the orientation. In the Practice panel a random system appears and you decide whether it is right- or left-handed before the hand reveals it.
Frequently asked questions
- How do you recognise a right-handed system?
- With the determinant: (a, b, c) is right-handed if det(a, b, c) = a · (b × c) is positive. Without calculation, use the right-hand rule: thumb a, index finger b, and the middle finger must point to the side of c.
- What happens when you swap two vectors?
- The orientation reverses: a right-handed system becomes left-handed and the determinant changes sign. A cyclic shift is two swaps and keeps the orientation.
- What is a right-handed system in the plane?
- A pair (a, b) with det(a, b) = a₁b₂ − a₂b₁ > 0. Then a turns the shorter way counterclockwise to b, and the thumb of the right hand points out of the plane towards the viewer.
- What can I try in the app?
- Drag the arrow tips, switch between plane and space, apply the right or left hand, rotate, swap, reflect or cyclically shift the system, and identify right- and left-handed systems in the quiz.
Subject: Analytic geometry
More from Analytic geometry
- Point-to-line distance in 3D
- Distance between skew lines in 3D
- Planes in space
- Cross and scalar triple products in 3D
- Angles between vectors in 3D
- Angles between lines and planes in 3D
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