CLASSICAL-MECHANICS

Multiplying Vectors Into a New Dimension: The Cross Product

IKey Formulas

Formula	Name	Description
$\lvert \mathbf{A}\times\mathbf{B}\rvert = \lvert\mathbf{A}\rvert\,\lvert\mathbf{B}\rvert\sin\theta$	Magnitude	Equals area of the parallelogram spanned by $\mathbf{A}$ and $\mathbf{B}$ ( $0 \le \theta \le \pi$ )
Right-hand rule	Direction	Curl fingers from $\mathbf{A}$ toward $\mathbf{B}$ ; thumb gives $\mathbf{A}\times\mathbf{B}$ , perpendicular to their plane
$\mathbf{A}\times\mathbf{B} = -\,\mathbf{B}\times\mathbf{A}$	Anti-commutativity	Swapping order flips the sign
$(\mathbf{A}+\mathbf{B})\times\mathbf{C} = \mathbf{A}\times\mathbf{C} + \mathbf{B}\times\mathbf{C}$	Distributive	Scalars factor out; product distributes over sums

IIInteractive Visualizations

Visualization 1 — Parallelogram Area

The magnitude $\lvert\mathbf{A}\times\mathbf{B}\rvert$ is the area of the parallelogram the two vectors span.

|A| 3 |B| 4 θ 90°

💡 Challenge: double just $\lvert\mathbf{A}\rvert$ and check whether the area doubles too.

Visualization 2 — Direction & the Right-Hand Rule

The product points perpendicular to the plane, and reversing the order flips it to the opposite side.

Compute θ 60°

Visualization 3 — Magnitude Versus Angle

Because the magnitude follows $\sin\theta$ , it peaks at $90^\circ$ and vanishes when the vectors line up ( $\lvert\mathbf{A}\rvert = \lvert\mathbf{B}\rvert = 1$ ).

θ 90°

💡 The dot product (dashed) follows $\cos\theta$ — it peaks exactly where the cross product vanishes.

IIIQuiz Questions

Question 1

Two vectors satisfy $\lvert\mathbf{A}\rvert = 3$ and $\lvert\mathbf{B}\rvert = 4$ , and the angle between them is $\theta = 90°$ .

What is the magnitude $\lvert\mathbf{A}\times\mathbf{B}\rvert$ ?

✅ Correct! At 90°, sin θ = 1, so the magnitude is 3·4·1 = 12.

❌ Not quite. Apply |A×B| = |A||B|sin θ with sin 90° = 1.

Show solution

Solution:

Use the magnitude formula: $\lvert\mathbf{A}\times\mathbf{B}\rvert = \lvert\mathbf{A}\rvert\,\lvert\mathbf{B}\rvert\sin\theta = 3 \cdot 4 \cdot \sin 90° = 12 \cdot 1 = 12.$

At $\theta = 90°$ , $\sin\theta = 1$ , which gives the maximum possible magnitude for these lengths — equal to the area of the rectangle they span.

Question 2

A vector $\mathbf{A}$ is crossed with another vector that points in the same direction (the two vectors are parallel, $\theta = 0$ ).

What is $\lvert\mathbf{A}\times\mathbf{B}\rvert$ ?

✅ Correct! Parallel vectors give sin 0 = 0, so the cross product vanishes.

❌ Not quite. The cross product uses sin θ, not cos θ. With θ = 0, sin 0 = 0.

Show solution

Solution:

For parallel vectors $\theta = 0$ , and $\sin 0 = 0$ , so $\lvert\mathbf{A}\times\mathbf{B}\rvert = \lvert\mathbf{A}\rvert\,\lvert\mathbf{B}\rvert\sin 0 = 0.$

Geometrically the parallelogram collapses to a line with zero area. The common mistake is to use $\cos\theta$ (that's the dot product) — the cross product uses $\sin\theta$ , which is zero here. In particular, a vector crossed with itself is always zero.

Question 3

True or False: For any two vectors, $\mathbf{A}\times\mathbf{B} = \mathbf{B}\times\mathbf{A}$ (the cross product is commutative, just like the dot product).

✅ Correct! The cross product is anti-commutative: A×B = −B×A.

❌ Not quite. Swapping the order flips the direction, so A×B = −B×A.

Show solution

Solution:

The statement is False. The cross product is anti-commutative: $\mathbf{A}\times\mathbf{B} = -\,\mathbf{B}\times\mathbf{A}.$

Reversing the order makes the right-hand rule arc sweep the opposite way, so your thumb flips to the opposite direction. The magnitude stays the same, but the direction reverses. This is a sharp contrast with the dot product, which is commutative.

Question 4

Given that $\mathbf{A}\times\mathbf{B} = \mathbf{C}$ , what is the value of the expression $(2\mathbf{A})\times\mathbf{B} + \mathbf{B}\times\mathbf{A}$ ?

Hint: use the scalar rule and anti-commutativity.

✅ Correct! 2C + (−C) = C, combining the scalar rule and anti-commutativity.

❌ Not quite. (2A)×B = 2C and B×A = −C, so the sum is 2C − C = C.

Show solution

Solution:

Apply the algebraic properties term by term.

Scalar rule: $(2\mathbf{A})\times\mathbf{B} = 2(\mathbf{A}\times\mathbf{B}) = 2\mathbf{C}.$

Anti-commutativity: $\mathbf{B}\times\mathbf{A} = -(\mathbf{A}\times\mathbf{B}) = -\mathbf{C}.$

Add them: $2\mathbf{C} + (-\mathbf{C}) = \mathbf{C}.$

So the expression equals $\mathbf{C}$ .

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