Vectors u and v are both nonzero, and their dot product u·v = 0. A student concludes at least one must actually be zero, reasoning 'you can't multiply two nonzero numbers and get zero.' What error is this?
AThe student is correct — zero dot product requires at least one zero vector
BThe dot product is not scalar multiplication of lengths; the formula u·v = |u||v|cos(θ) shows that two nonzero vectors can have zero dot product when θ = 90°, since cos(90°) = 0
CThe student should check whether the vectors are in the same dimension before concluding
DZero dot product means the vectors are parallel, not that one must be zero
The dot product is not a simple multiplication of magnitudes — it includes the cosine of the angle between them. When two nonzero vectors are perpendicular, cos(90°) = 0, making the entire product zero regardless of the magnitudes. This is precisely the orthogonality test: u·v = 0 if and only if u and v are perpendicular (or one is the zero vector). The student's error was treating the dot product like regular scalar multiplication.
Question 2 Multiple Choice
The vector projection proj_a(b) represents which geometric quantity?
AThe component of b that is perpendicular to a
BA vector in the direction of a whose length equals the full magnitude of b
CThe component of b along the direction of a — its shadow cast onto the line through a
DThe vector midpoint between a and b
proj_a(b) decomposes b into its component parallel to a. The formula proj_a(b) = ((a·b)/|a|²)a gives a vector in the direction of a, scaled by how much of b lies in that direction. The remaining component b − proj_a(b) is perpendicular to a by construction. This decomposition — parallel plus perpendicular — is one of the most repeatedly useful tools in vector calculus, underlying everything from distance-to-a-plane computations to directional derivatives.
Question 3 True / False
Two vectors with equal magnitudes is expected to have a non-negative dot product.
TTrue
FFalse
Answer: False
Equal magnitudes say nothing about direction. If two vectors of equal length point in opposite directions (θ = 180°), their dot product is |u||v|cos(180°) = −|u|², which is negative. The dot product depends on both magnitude and the cosine of the angle. Equal magnitudes fix |u||v| but leave cos(θ) free to be positive, zero, or negative.
Question 4 True / False
The scalar projection of b onto a gives the signed length of b's component in the direction of a.
TTrue
FFalse
Answer: True
The scalar projection comp_a(b) = (a·b)/|a| = |b|cos(θ) is the signed length of b's shadow along a. It is positive when b has a component in the same direction as a (θ < 90°), zero when perpendicular, and negative when b leans away from a (θ > 90°). The signed quality is important: it tells you not just how long the projection is but which way it points along a.
Question 5 Short Answer
Why does the gradient vector ∇f point in the direction of steepest ascent? Use the dot product to explain.
Think about your answer, then reveal below.
Model answer: The directional derivative of f in direction u is ∇f · u. By the formula a·b = |a||b|cos(θ), this equals |∇f||u|cos(θ). Since u is a unit vector (|u| = 1), this simplifies to |∇f|cos(θ), which is maximized when cos(θ) = 1 — i.e., when u aligns exactly with ∇f. The gradient points in the direction that maximizes alignment with itself, which is by definition the direction of steepest ascent.
This is why learning the dot product as a measure of alignment pays off immediately in vector calculus. Every question about 'how fast does f change in direction u?' reduces to a dot product with the gradient. The direction of steepest ascent is the direction that maximizes cos(θ) — a purely geometric fact about alignment that follows directly from the cosine interpretation of the dot product.