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4.5.1 · D5Linear Algebra (Full)

Question bank — Vectors in ℝⁿ — operations, geometric interpretation

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True or false — justify

True or false: for all vectors, but too.
The first is true (addition is component-wise and real addition commutes); the second is false — , so they are equal only when .
True or false: the dot product of a vector with itself can be negative.
False — is a sum of squares, so it is always , and equals only for the zero vector.
True or false: if then .
False — equal length only means the arrows are equally long; they can point in completely different directions, e.g. and .
True or false: scaling a vector by always makes it longer.
False — the new length is , so shrinks it, collapses it to , and also flips its direction.
True or false: means at least one of is the zero vector.
False — it means they are perpendicular; two perfectly good nonzero arrows like and have dot product .
True or false: normalizing a vector changes the angle it makes with the axes.
False — dividing by a positive length rescales every component by the same factor, preserving all ratios and hence all directions and angles.
True or false: for any two vectors.
False in general — equality holds only when they point the same way; otherwise the tip-to-tail arrow takes a "shortcut," so (the triangle inequality).
True or false: the dot product is a vector because both inputs are vectors.
False — the dot product outputs a single scalar; it measures alignment rather than pointing anywhere. See Dot Product and Orthogonality.

Spot the error

A student writes "." Where is the mistake?
They kept the components separate instead of summing them; the dot product is , one number, not a pair.
A student computes . What went wrong?
They forgot the Pythagorean structure — length is ; simply adding components can even give the wrong sign or zero for a nonzero vector.
"To normalize , divide by its length ." Fix it.
You cannot — the zero vector has length and dividing by is undefined; the zero vector simply has no direction to normalize.
"Since , I got ." What's impossible here?
can never exceed ; a value signals an arithmetic slip — the Cauchy–Schwarz inequality guarantees .
" where : I doubled to then wrote the answer as ." Find the flaw.
You subtracted a scalar from a vector; must be subtracted component-wise: .
", so if the dot product is ." Spot the slip.
, not ; anti-parallel vectors give the most negative dot product . The zero case is .

Why questions

Why is vector addition defined component-wise and not some other way?
Because each axis is independent — moving east never changes your northward position — so the only consistent rule combines matching coordinates separately.
Why does equal rather than ?
The dot product multiplies matching components and sums, giving , which is exactly what sits inside the square root of the norm — so it is the squared length.
Why does the Law of Cosines derivation make the and terms cancel?
Both the geometric expansion and the algebraic dot-product expansion of contain those two terms identically, so setting the expressions equal removes them, isolating .
Why can't we always define a cross product in the way we do in ?
A cross product needs a unique direction perpendicular to two vectors, which only provides; in higher dimensions the perpendicular space has many directions. See Cross Product (R3 only).
Why does normalizing use division by the norm and not subtraction?
We want to keep the direction but rescale the length to ; multiplying every component by shrinks uniformly, whereas subtracting would shift the arrow and change its direction.
Why is the dot product the natural tool for detecting perpendicularity rather than comparing components directly?
Because isolates the angle: the lengths are always positive, so the sign and zeros of the dot product report exactly the geometry of .

Edge cases

What is the dot product of the zero vector with any vector?
Always , since every term — but this does not mean is "perpendicular"; has no direction, so the angle is undefined.
What angle does the dot-product formula assign when one vector is ?
None — is , undefined, matching the fact that a zero-length arrow has no direction to measure against.
For (a vector with itself), what does the angle formula give?
, so — a vector is perfectly aligned with itself, as expected.
For , what angle and dot product result?
and , giving — the arrows point in exactly opposite directions.
In , what does the "dot product" reduce to, and what does its sign tell you?
It is just the ordinary product ; its sign tells you whether the two numbers point the same way (positive) or opposite ways (negative) on the number line — the same-direction / opposite-direction idea in miniature.
What is in terms of and , and why the absolute value?
; length is never negative, so a negative scalar flips direction but its magnitude sets the stretch.
Can two different nonzero vectors have the same normalized form ?
Yes — any positive scalar multiples of the same vector, like and , share one unit vector because normalizing erases length and keeps only direction.
Recall One-line summary of the traps

Most errors here come from three confusions: treating the dot product as a vector, forgetting the square root (or the squares) in the norm, and forgetting that flips direction while sets the stretch. Guard those three and the edge cases fall into place.

Connections