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Mathematics · Ch 8 — Vectors

Vector (Cross) Product of Vectors

8

Vector (Cross) Product of Vectors

Geometric Definition

For two nonzero, non-parallel vectors a⃗\vec a and b⃗\vec b with angle θ\theta between them (0≤θ≤π0\le\theta\le\pi), the vector product (or cross product) a⃗×b⃗\vec a\times\vec b is itself a vector, defined by

a⃗×b⃗=(∣a⃗∣ ∣b⃗∣sin⁡θ)n^\vec a\times\vec b=\left(|\vec a|\,|\vec b|\sin\theta\right)\hat n

where n^\hat n is the unit vector perpendicular to both a⃗\vec a and b⃗\vec b, chosen so that a⃗,b⃗,n^\vec a,\vec b,\hat n form a right-handed system -- found by the right-hand rule: curl the fingers of the right hand from a⃗\vec a towards b⃗\vec b through the angle θ\theta; the thumb then points along n^\hat n. If a⃗\vec a and b⃗\vec b are parallel, or either is 0⃗\vec 0, then a⃗×b⃗=0⃗\vec a\times\vec b=\vec 0.

Key Consequences of the Definition

  • Not commutative: b⃗×a⃗=−(a⃗×b⃗)\vec b\times\vec a=-(\vec a\times\vec b) -- reversing the order reverses the direction of n^\hat n (curl the fingers the other way), though the magnitude is unchanged.
  • Parallelism test: since sin⁡0=sin⁡π=0\sin0=\sin\pi=0, a⃗×b⃗=0⃗  ⟺  a⃗∥b⃗\vec a\times\vec b=\vec 0\iff\vec a\parallel\vec b (for nonzero a⃗,b⃗\vec a,\vec b) -- the vector-product analogue of the dot product's perpendicularity test.
  • Magnitude equals area: ∣a⃗×b⃗∣=∣a⃗∣∣b⃗∣sin⁡θ|\vec a\times\vec b|=|\vec a||\vec b|\sin\theta is exactly the area of the parallelogram with adjacent sides a⃗\vec a and b⃗\vec b (base ×\times height, with height =∣b⃗∣sin⁡θ=|\vec b|\sin\theta).
  • Axis unit vectors: i^×i^=j^×j^=k^×k^=0⃗\hat i\times\hat i=\hat j\times\hat j=\hat k\times\hat k=\vec 0, and, applying the right-hand rule around the cyclic order i→j→k→ii\to j\to k\to i: i^×j^=k^\hat i\times\hat j=\hat k, j^×k^=i^\hat j\times\hat k=\hat i, k^×i^=j^\hat k\times\hat i=\hat j (with j^×i^=−k^\hat j\times\hat i=-\hat k, etc., for the reverse order).
  • Distributivity: a⃗×(b⃗+c⃗)=a⃗×b⃗+a⃗×c⃗\vec a\times(\vec b+\vec c)=\vec a\times\vec b+\vec a\times\vec c.

Component (Determinant) Formula

Let a⃗=a1i^+a2j^+a3k^\vec a=a_1\hat i+a_2\hat j+a_3\hat k and b⃗=b1i^+b2j^+b3k^\vec b=b_1\hat i+b_2\hat j+b_3\hat k. Expanding a⃗×b⃗\vec a\times\vec b term by term using the i^,j^,k^\hat i,\hat j,\hat k cross-product rules above and collecting like terms gives the compact determinant form:

a⃗×b⃗=∣i^j^k^a1a2a3b1b2b3∣=(a2b3−a3b2)i^−(a1b3−a3b1)j^+(a1b2−a2b1)k^\vec a\times\vec b=\begin{vmatrix}\hat i&\hat j&\hat k\\ a_1&a_2&a_3\\ b_1&b_2&b_3\end{vmatrix}=(a_2b_3-a_3b_2)\hat i-(a_1b_3-a_3b_1)\hat j+(a_1b_2-a_2b_1)\hat k

Important

The middle (j^\hat j) term carries a minus sign from the determinant expansion -- this is the single most common sign slip when computing a cross product by hand; always expand along the top row and alternate +,−,++,-,+.

Worked Application: Area of a Triangle …

Figure 1Cross product, the right-hand rule and area

What this figure shows. Two vectors labelled a and b are drawn from a common initial point O, lying in a shaded plane, with the angle theta marked between them. A parallelogram is completed using a and b as adjacent sides, its interior shaded to represent the area |a||b|sin(theta), which equals the magnitude of the cross product. A third vector, labelled a x b (or n-hat), is drawn perpendicular to the shaded plane, with a small right-hand inset beside it showing the fingers curling from a toward b and the thumb pointing along the same perpendicular direction, illustrating the right-hand rule that fixes the cross product's direction; the perpendicul …

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