1.1.11 · D5Measurement, Vectors & Kinematics
Question bank — Dot product — formula, geometric meaning, work calculation
True or false — justify
is a vector because you started with two vectors.
False — the dot product collapses to a single scalar; direction is thrown away. The operation that keeps a direction is the cross product.
If then at least one of the vectors must be the zero vector.
False — two non-zero vectors give zero whenever they are perpendicular (). Zero dot product means "no alignment," not "no length."
can be negative if points into the third quadrant.
False — , a length squared, so it is always regardless of which way points.
Swapping the order, , can change the sign of the result.
False — the dot product is commutative; is the same whether you measure the angle from to or to , so .
A force perpendicular to the motion still does a little work because it is a real, strong force.
False — work is , and , so a perpendicular force does exactly zero work no matter how large it is (see Circular motion).
Making both vectors twice as long leaves between them unchanged.
True — scaling changes and the dot product, but in the extra factors cancel top and bottom, so the angle is unchanged.
can be larger than if the vectors are strongly aligned.
False — the biggest can be is , so always. Getting a bigger number means you made an arithmetic slip.
Distributivity, , only works when the vectors are perpendicular.
False — it holds for any vectors, because projections add: the shadow of on is the sum of the two separate shadows.
Spot the error
", so I plug in the dot product and take ."
Missing denominator — the correct relation is . Forgetting to divide by the magnitudes can give a "" bigger than 1, which is impossible.
"Work is force times distance, so for a rope at ."
Only the aligned part of the force does work. You must include : . Plain is the special case only.
"Friction did of work — that's an error, work can't be negative."
Not an error — negative work means energy was removed from the object (friction, braking). The minus sign is physical, coming from when the force opposes motion.
" because both are unit vectors of length 1."
The magnitudes are 1, but they are perpendicular, so . Matched pairs give 1; mixed pairs give 0 — that is the whole trick behind the component formula.
"."
The formula is — you add the products; the minus here came from , not from a subtraction rule. Sign lives inside the component, never in the operation.
"The dot product points in the direction halfway between and ."
A scalar has no direction at all. There is nothing to point anywhere — you're confusing it with a vector operation like the cross product.
Why questions
Why does — and not — appear in the dot product?
Because the dot product measures how much of one vector lies along the other (the projection, or shadow), and that aligned length is . The part is the perpendicular piece, which the cross product uses.
Why is the dot product perfect for computing work?
Only the component of force along the displacement moves an object, and that component is — exactly what builds in automatically (see Work-Energy Theorem).
Why do all the "mixed" terms vanish when you expand the component form?
Each mixed term carries a factor like , so only like-with-like products () survive, leaving .
Why does give and not just ?
A vector dotted with itself has , so , giving . It's magnitude times magnitude, hence squared — this is how the length formula falls out.
Why can two definitions — the geometric and the component — always agree?
They're proven equal: writing each vector in the perpendicular unit-vector basis and expanding with distributivity kills every mixed term, converting the projection picture into the component sum (see Trigonometry — cosine and components).
Why does the sign of alone tell you if the angle is sharp or wide?
Because , so the sign comes purely from : ==positive for == (partly aligned), zero at , negative for (opposing).
Edge cases
What is when is the zero vector?
Zero — the zero vector has no length and no direction, so its shadow on anything is nothing. (Note: the angle is undefined here, so use the component form to be safe.)
Two vectors are antiparallel (point exactly opposite, ). What is their dot product?
The most negative it can be: . This is the sign you get for friction directly opposing motion.
A puck slides in a circle at constant speed; what work does the centripetal force do per lap?
Zero — the centripetal force always points perpendicular to the velocity, so at every instant, giving zero work and unchanged speed (see Circular motion).
If is exactly but the vectors are enormous, is the dot product large?
No — magnitude doesn't matter when ; the product is zero however long the vectors are. Perpendicularity beats size.
What does a maximum dot product tell you about the vectors?
They are parallel and same direction (, ). Every bit of lies along , so the shadow equals the full length.
Can be positive while the vectors point in visibly different directions?
Yes — any angle between and gives a positive (but not maximal) result; the vectors are partly aligned, not identical. Positive just means "sharp angle."
If you only have components and no angle, can you still find the work?
Yes — use directly; the component form needs no angle at all, which is why it's often the faster route.
Recall One-line self-test
Cover the answers. If you can justify every "True/False" with a reason (not a coin-flip), and you can name why beats here, you own this topic. Ready ::: Then move on to the cross product — the "" partner that keeps a direction.