Shoelace method: the cross product
About this app
The cross product a × b by the school rule: two columns, a copy below, cross out the first and last row, lace three times crosswise (down minus back). The laces draw themselves, the calculations stand next to them, the result gets brackets. Check with the dot product using a, b or c = (7, 8, 9), |a × b| as the area of the parallelogram.
With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.
The school rule for a × b
The cross product of two vectors in space has the components a2b3 − a3b2, a3b1 − a1b3 and a1b2 − a2b1. The shoelace method turns this into a recipe without memorising: write a and b without brackets as two columns, copy both columns straight below, cross out the first and last row. In the four middle rows each cross of two neighbouring rows gives one component: product down minus product back. The middle component comes out with the right sign as well.
What the app shows
The laces draw themselves between the numbers, down and back in two colours, and the calculation appears next to each cross. For (1, 2, 3) × (4, 5, 6): 2 · 6 − 3 · 5 = −3, 3 · 4 − 1 · 6 = 6, 1 · 5 − 2 · 4 = −3. The three numbers glide into vector brackets: a × b = (−3, 6, −3). At the end the laces pull tight like a bow.
Try it and check
The components of a and b can be changed between −9 and 9, and there are examples: basis vectors e1 × e2 = e3, parallel vectors giving the zero vector and an exam problem. The check computes the dot product with a, with b or with a freely chosen c. With c = (7, 8, 9) the result is 0 as well, because the keypad rows 1-2-3, 4-5-6, 7-8-9 of a phone lie in one plane. The length |a × b| is shown in the panel as the area of the parallelogram.
Frequently asked questions
- How do you calculate the cross product of two vectors?
- Component by component: a × b = (a2b3 − a3b2, a3b1 − a1b3, a1b2 − a2b1). With the shoelace method you write a and b as columns, copy them below, cross out the first and last row and compute three times crosswise in the four middle rows: down minus back.
- How do you check a cross product?
- a × b is perpendicular to a and to b, so the dot product with a and with b must be 0. Example: (−3, 6, −3) · (1, 2, 3) = −3 + 12 − 9 = 0.
- What does the length of the cross product mean?
- |a × b| is the area of the parallelogram spanned by a and b. For (1, 2, 3) and (4, 5, 6) it is √54 ≈ 7.35. For parallel vectors the area is 0 and a × b is the zero vector.
- Can I enter my own vectors in the app?
- Yes. All six components of a and b can be set between −9 and 9; laces, calculations, result and check update at once. You can go step by step or play the whole sequence.
Subject: Analytic geometry
More from Analytic geometry
- Point-to-line distance in 3D
- Distance between skew lines in 3D
- Planes in space
- Right-handed systems in 2D and 3D
- Dot product: coordinate formula from the definition
- Cross and scalar triple products in 3D
- Angles between vectors in 3D
- Angles between lines and planes in 3D
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