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Linear Algebra 10 | Cross Product thumbnail

Linear Algebra 10 | Cross Product

The Bright Side of Mathematics·
3 min read

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TL;DR

The cross product u  v is defined for vectors in 3 and returns a vector perpendicular to both u and v.

Briefing

The cross product is a uniquely three-dimensional operation: given two vectors in 3, it produces a third vector that is perpendicular to both inputs. That perpendicular “direction” and the vector’s “length” are not just algebraic artifacts—they connect directly to geometry, letting you compute areas and build orthogonal vectors.

For vectors u and v in 3, the cross product is written as u  v and yields another vector. In component form, if u = (u1, u2, u3) and v = (v1, v2, v3), then u  v = (u2v3 − u3v2, u3v1 − u1v3, u1v2 − u2v1). The transcript emphasizes a memory-friendly pattern: start in the middle, multiply “downward” pairs, mirror across the diagonal, and subtract the mirrored products to get each component.

An alternative expression uses the Levi-Civita symbol (epsilon) to write the cross product as a sum over indices: (u  v)k = 

Cornell Notes

In 3, the cross product u  v takes two 3D vectors and returns a third vector perpendicular to both. Its components follow a structured subtraction pattern: each component is “downward” products minus the mirrored “downward” products. Three key geometric properties follow: (1) orthogonality, meaning (u  v)  u = 0 and (u  v)  v = 0 under the standard inner product; (2) direction given by the right-hand rule; and (3) magnitude equal to the area of the parallelogram spanned by u and v (so a triangle’s area is half). The operation is order-sensitive, so u  v generally differs from v  u.

Why does the cross product produce a vector orthogonal to both inputs?

For vectors u and v in 3, the cross product u  v is constructed so that its dot product with u and with v is zero under the standard inner product. In practice, once the component formula (u2v3 − u3v2, u3v1 − u1v3, u1v2 − u2v1) is used, the resulting vector’s components cancel when dotted with either original vector, confirming orthogonality.

How can the direction of u  v be determined without recomputing components?

The right-hand rule provides the direction. Point the thumb along u, the index finger along v, and lift the middle finger perpendicular to both; the middle finger indicates the direction of u  v. Reversing the order of u and v flips the direction.

What does the length |u  v| represent geometrically?

The magnitude of u  v equals the area of the parallelogram formed by u and v. That means if u and v span a parallelogram, the cross product’s length matches its area; for a triangle formed by the same two vectors, the area is half of that value.

What is the component formula for u  v in 3?

If u = (u1, u2, u3) and v = (v1, v2, v3), then u  v = (u2v3 − u3v2, u3v1 − u1v3, u1v2 − u2v1). The transcript highlights a memory method: start in the middle, multiply downward pairs, mirror, and subtract to get each component.

How does the cross product behave under swapping the order of vectors?

It is not commutative. The right-hand rule depends on the order: u  v points one way, while v  u points the opposite way (unless the cross product is the zero vector).

Review Questions

  1. Given u = (a, b, c) and v = (d, e, f), write the three components of u  v.
  2. Using the right-hand rule, what happens to the direction of u  v if u and v are swapped?
  3. If |u  v| = 12, what is the area of the parallelogram and the area of the triangle formed by u and v?

Key Points

  1. 1

    The cross product u  v is defined for vectors in 3 and returns a vector perpendicular to both u and v.

  2. 2

    For u = (u1, u2, u3) and v = (v1, v2, v3), u  v = (u2v3 − u3v2, u3v1 − u1v3, u1v2 − u2v1).

  3. 3

    Orthogonality means (u  v)  u = 0 and (u  v)  v = 0 under the standard inner product.

  4. 4

    The right-hand rule determines the direction of u  v: thumb along u, index along v, middle gives u  v.

  5. 5

    The magnitude |u  v| equals the area of the parallelogram spanned by u and v; triangle area is half.

  6. 6

    The cross product is not commutative: u  v generally differs from v  u.

  7. 7

    A concrete example with u = (2, 1, 0) and v = (0, 1, 0) yields u  v = (0, 0, 2), pointing along the z-axis.

Highlights

u  v is perpendicular to both u and v, turning algebra into a reliable way to find orthogonal directions.
The right-hand rule links vector order to direction: swapping u and v flips the resulting vector.
|u  v| equals the parallelogram area spanned by u and v, making the cross product a direct geometry tool.

Topics

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