Cross Product, Area, and Scalar Triple Product
A cross B has magnitude AB sin(theta) and right-hand-rule direction. The scalar triple product A dot (B cross C) is zero exactly when the three vectors are coplanar, apart from degenerate cases.
Why this shows up in the exam
Finding normals to planes · Computing torque and angular momentum · Testing coplanarity and volumes
Learn the idea
The cross product produces a perpendicular vector whose magnitude measures oriented area. Two nonparallel arrows span a parallelogram. Their cross product points normal to that plane by the right-hand rule, while a scalar triple product measures signed volume and tests coplanarity.
🧠 Memory hook: Cross means across: the answer points out of the two-vector plane.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- |A cross B| = AB sin(theta) — magnitude of the cross product
- A dot (B cross C) = 0 — coplanarity condition
- tau_vector = r_vector cross F_vector — torque about the chosen origin
How to approach it
- 1Place vectors in a common basis
- 2Evaluate the determinant with order preserved
- 3Check direction or coplanarity as the question requires
Common slip-ups that cost marks
- •Using the left-hand rule
- •Forgetting that reversing order changes sign
- •Treating a zero cross product as proof that both vectors are zero
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
A particle has initial speed 2 m/s and constant acceleration 3 m/s^2 for 4 s. What distance does it cover?
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