The cross product is a vector operation between two vectors A and B that produces a new vector with magnitude AB sin θ, pointing perpendicular to both original vectors (normal to the plane they define). In AP Physics C: Mechanics, it's the math behind torque, τ = r × F, and angular momentum, L = r × p.
The cross product is one of two ways to multiply vectors, and unlike the dot product, the answer is itself a vector. Take two vectors A and B with an angle θ between them. The cross product A × B has magnitude AB sin θ and points perpendicular to both A and B, sticking straight out of the plane the two vectors lie in. You find the direction with the right-hand rule: point your fingers along A, curl them toward B, and your thumb gives the direction of A × B.
Two properties matter most for the AP exam. First, the sin θ factor means the cross product only "sees" the perpendicular component of the vectors. If A and B are parallel (θ = 0°), the cross product is zero. If they're perpendicular (θ = 90°), it's at its maximum, AB. Second, order matters. A × B = −(B × A), so swapping the vectors flips the resulting vector's direction. That's why torque is always written r × F, position vector first.
The cross product is the mathematical engine of Topic 5.3 (Torque) in Unit 5, where torque is defined as τ = r × F, the cross product of the position vector from the pivot to the point of application and the applied force. The sin θ in the magnitude is exactly why a force applied parallel to the lever arm produces zero torque and a perpendicular force produces maximum torque. It also explains why torque, angular velocity, and angular momentum point along rotation axes instead of along the motion itself. Once you're comfortable with the cross product in Unit 5, the angular momentum definition L = r × p in the rotational energy and momentum unit is the same operation with momentum swapped in for force. If you can compute |r||F| sin θ and apply the right-hand rule, you've unlocked most of rotational dynamics.
Keep studying AP® Physics C: Mechanics Unit 5
Torque (Unit 5)
Torque is the cross product in action: τ = r × F. The magnitude rF sin θ tells you only the perpendicular component of the force twists the object, and the right-hand rule tells you whether the torque vector points into or out of the page.
Lever arm / moment arm (Unit 5)
The lever arm is the cross product hiding in disguise. Writing torque as F times the moment arm (r sin θ) is the same calculation as |r × F|, just regrouped. They're two notations for one idea.
Angular momentum (Unit 6)
Angular momentum of a particle is L = r × p, structurally identical to torque. Master the cross product once in Unit 5 and the Unit 6 definition is a free transfer, including why a particle moving straight past a point still has angular momentum about it.
Position vector (Units 1 and 5)
The r in r × F is the position vector from the pivot to where the force acts. Choosing where r starts (your pivot choice) changes the torque you calculate, which is the whole strategy behind smart pivot selection in static equilibrium problems.
Cross product questions in Mechanics almost always wear a torque costume. Multiple-choice stems ask things like which operation produces a vector perpendicular to both the force and position vectors (answer: the cross product), or which method determines whether a torque vector points into or out of the page (answer: the right-hand rule). Calculation problems, like finding the net torque from several forces acting on the rim of a disk at different angles, require you to compute rF sin θ for each force and assign signs based on rotation direction. On FRQs, you won't be asked to define the cross product, but you'll use it constantly: setting up Στ = Iα, justifying why a force through the pivot contributes zero torque (r and F are parallel or r = 0), and arguing direction with the right-hand rule. Knowing that maximum torque happens at θ = 90° is a fast win on conceptual questions.
Both multiply two vectors, but they answer different questions. The dot product (A · B = AB cos θ) gives a scalar and measures how much the vectors align; it shows up in work, W = F · d. The cross product (|A × B| = AB sin θ) gives a vector and measures how much the vectors are perpendicular; it shows up in torque, τ = r × F. Quick check: parallel vectors maximize the dot product but make the cross product zero, and it's the exact opposite at 90°.
The cross product A × B is a vector with magnitude AB sin θ that points perpendicular to both A and B, normal to the plane they define.
Use the right-hand rule for direction: fingers point along the first vector, curl toward the second, and your thumb gives the result (this is how you decide if torque points into or out of the page).
Order matters because A × B = −(B × A), which is why torque must be written as r × F with the position vector first.
The cross product of parallel vectors is zero, so a force directed along the line to the pivot produces no torque.
Torque (τ = r × F) and angular momentum (L = r × p) are both cross products, so learning the operation once pays off in Unit 5 and Unit 6.
The cross product hits its maximum value AB when the two vectors are perpendicular, which is why torque is maximized when force is applied at 90° to the lever arm.
It's a vector multiplication where A × B has magnitude AB sin θ and points perpendicular to both A and B. On the exam it appears mainly in torque (τ = r × F) and angular momentum (L = r × p).
The dot product gives a scalar (AB cos θ) and is maximized for parallel vectors, like in work, W = F · d. The cross product gives a vector (magnitude AB sin θ) and is maximized for perpendicular vectors, like in torque.
Yes. With θ = 0°, sin θ = 0, so the cross product vanishes. Physically, that's why a force pointing directly toward or away from a pivot produces zero torque.
Use the right-hand rule: point your fingers along the first vector (r for torque), curl them toward the second vector (F), and your thumb points in the direction of r × F. This is the standard exam method for deciding whether a torque vector points into or out of the page.
Yes. The cross product is anticommutative, meaning A × B = −(B × A). Swapping the order flips the direction of the result, so torque is always r × F, never F × r.
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