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1.1.12 · D5Measurement, Vectors & Kinematics

Question bank — Cross product — formula, direction (right-hand rule), torque - area calculation

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Before the traps, one shared picture to point at. Two arrows laid tail-to-tail sweep out a slanted patch; the cross product's length is that patch's area, and its direction is the skewer standing straight up out of the patch.

Figure — Cross product — formula, direction (right-hand rule), torque - area calculation

True or false — justify

Each item: decide true/false, then give the one-sentence reason. The reason is the point.

is a scalar because it measures an area.
False. It encodes an area in its magnitude, but the whole object is a vector — the area comes with a perpendicular direction (the "which way does the patch face" skewer), and that direction is essential.
If then at least one of the vectors is the zero vector.
False. It is also zero when both are nonzero but parallel or antiparallel ( or , ) — two arrows on the same line span a patch of zero width.
always points in a direction perpendicular to both and .
True. It is perpendicular to the entire plane they span, so it is perpendicular to every vector lying in that plane, including and themselves.
Swapping the order, , gives a vector of different length.
False. Same length ( is unchanged), opposite direction — the right-hand curl reverses, so the thumb flips: .
for every vector .
True. A vector is parallel to itself, , so and there is no patch — a line has no area.
The cross product is largest when the two vectors are most aligned.
False. It is largest when they are perpendicular (, ). Alignment is what the dot product rewards; the cross product rewards spread.
always.
True. is perpendicular to , and the dot product of perpendicular vectors is zero — this is a fast sanity check on any computed cross product.
The cross product obeys the associative law: .
False. Cross product is not associative; the two sides generally point differently because the parenthesised pair defines a different plane each time.

Spot the error

Read the reasoning, find the flawed step, and state the fix.

" is huge, so the torque must be huge."
The error ignores direction: if is parallel to (pushing straight along the lever), and no matter how large is. Only the perpendicular part turns things.
"To get the area of the triangle , I take using the origin ."
Wrong tails — the two edge vectors must share a vertex of the triangle, e.g. and from . Vectors from the unrelated point span the wrong parallelogram.
"Expanding the determinant left to right, the term is ."
Missing the cofactor minus sign. The pattern is , so the term is .
" because ."
Order matters — reversing flips the sign, so . The cycle is only going forward.
"The torque came out as , so the object rotates in the direction."
A torque is not a rotation direction; it names the axis. (out of the page) means counter-clockwise spin in the page by the right-hand rule — the arrow is the spin-axis, not the path of motion.
"Since uses and can be negative, the magnitude can be negative for obtuse angles."
For the angle between two vectors we always take , where , so the magnitude is never negative. The sign information lives in the direction, not the length.
"Both and describe the same parallelogram, so they are the same vector."
They describe the same patch (same area) but the skewer points to opposite faces of it — same magnitude, opposite direction, hence they differ by a minus sign.

Why questions

Answer the "why", not just the "what".

Why does the cross product return a vector while the dot product returns a number?
Because rotation and orientation need a direction in space (which way the axis points, which way a surface faces), and a plain number cannot carry that — a perpendicular vector can. See Dot product — formula, projection, work calculation for the scalar counterpart.
Why is the height of the parallelogram and not ?
Height is measured perpendicular to the base ; the perpendicular component of is , while is the part along , which adds nothing to the width.
Why does only the perpendicular part of a force create torque?
A force pointing along the lever arm just pushes toward or away from the pivot and cannot make it swing; only the sideways () component twists it — this is exactly the door-at-the-hinge picture. See Torque and rotational equilibrium.
Why does the determinant's middle term carry a minus sign?
The alternating cofactor pattern is the algebraic fingerprint of anti-commutativity; it guarantees that swapping the and rows flips the whole result's sign. Built the same way as in Determinants — 3×3 expansion.
Why is guaranteed perpendicular to the plane rather than just to one vector?
The magnetic-force and area interpretations demand a single unambiguous "facing" direction for the whole patch; being perpendicular to the plane makes it perpendicular to every vector in it at once. This is why Magnetic force — F = qv × B uses it.
Why do we say the cross product is "maximal when the dot product is zero"?
They use and , which are complementary: when vectors are perpendicular, (cross is max) and (dot is zero), and the reverse when parallel.
Why must the two vectors share a tail before you curl your right hand?
The angle and the plane are only well-defined once both arrows start from the same point; from a shared tail the sweep from to is unambiguous, fixing both magnitude and direction.

Edge cases

Boundary and degenerate inputs — the scenarios formulas quietly assume away.

What is when (parallel)?
The zero vector: gives zero magnitude, and there is no plane to define a direction — the "patch" has collapsed to a line.
What is when (antiparallel)?
Also the zero vector: . The arrows lie on one line pointing opposite ways, still spanning no area.
What happens if one input is the zero vector, e.g. ?
The result is — there is no arrow to span a patch with, so both magnitude () and direction are undefined-but-zero.
At exactly , how do magnitude and direction behave?
Magnitude hits its maximum (since ), and the direction is cleanly perpendicular to both — this is the "cleanest" cross product, the reference case.
If and are 3D but happen to both lie in the -plane, where does point?
Straight along (out of or into the plane), because the only direction perpendicular to the whole -plane is the -axis — the sign is set by the right-hand rule.
For angular momentum , what is when the particle moves straight toward the origin?
Zero, because and are then parallel (), so there is no rotational "spread". See Angular momentum — L = r × p.
Can two different pairs of vectors give the exact same cross product vector?
Yes — any pair spanning the same-area parallelogram with the same orientation in the same plane yields an identical result; the cross product does not uniquely determine its inputs.
Recall One-line self-test before you leave

Cross product: length is area of the parallelogram (), direction is the right-hand skewer perpendicular to the plane, and it vanishes for parallel vectors and peaks for perpendicular ones.

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