Chapter 2 Quiz — Vectors
Twelve quick conceptual checks. Try each before opening the answer. Each answer includes a one-line explanation.
Q1. A vector is best described as: (a) a point in space; (b) a list of numbers only; (c) a directed quantity with magnitude and direction, equivalently a list of components; (d) a grid of numbers.
Answer
**(c).** A vector is an arrow (magnitude + direction) and equivalently a list of components — the two views are the same object. A grid of numbers is a matrix (Chapter 7).Q2. An arrow from $(1, 1)$ to $(4, 5)$ represents which vector?
Answer
$\begin{bmatrix} 3 \\ 4 \end{bmatrix}$ (tip minus tail: $4-1=3$, $5-1=4$). Explanation: a vector is the *difference* of endpoints; position doesn't matter, only the displacement.Q3. True or false: $\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}$ for all vectors of the same dimension.
Answer
**True.** Vector addition is commutative — the parallelogram has the same far corner regardless of order, and componentwise it follows from commutativity of real-number addition.Q4. What does multiplying a vector by $-2$ do geometrically?
Answer
Reverses its direction *and* doubles its length. Explanation: the sign flips the direction; the magnitude $|-2| = 2$ scales the length by 2.Q5. Which sum is undefined: $\begin{bmatrix}1\\2\end{bmatrix} + \begin{bmatrix}3\\4\end{bmatrix}$ or $\begin{bmatrix}1\\2\end{bmatrix} + \begin{bmatrix}3\\4\\5\end{bmatrix}$?
Answer
The **second** — you can only add vectors of the same dimension. A 2-vector plus a 3-vector has no definition and no sensible picture.Q6. The magnitude of $\begin{bmatrix} 6 \\ 8 \end{bmatrix}$ is:
Answer
$\sqrt{36 + 64} = \sqrt{100} = 10$. Explanation: magnitude is the Pythagorean hypotenuse of the component legs.Q7. If $\lVert \mathbf{v} \rVert = 5$, what is $\lVert 3\mathbf{v} \rVert$? What is $\lVert -3\mathbf{v} \rVert$?
Answer
Both are **15**. Explanation: $\lVert c\mathbf{v}\rVert = |c|\,\lVert\mathbf{v}\rVert$, and $|3| = |-3| = 3$, so $3 \times 5 = 15$ in both cases — the sign doesn't affect length.Q8. A linear combination of vectors is:
Answer
A sum of scalar multiples, $c_1\mathbf{v}_1 + \cdots + c_k\mathbf{v}_k$. Explanation: "scale each, then add" — the single most important construction in the subject.Q9. What does the combination $(1-t)\mathbf{a} + t\mathbf{b}$ trace as $t$ goes from 0 to 1?
Answer
The straight segment from $\mathbf{a}$ (at $t=0$) to $\mathbf{b}$ (at $t=1$), passing through the midpoint at $t = \tfrac12$. This is linear interpolation ("lerp"); the weights sum to 1, which keeps you on the segment.Q10. In math the first component of $\mathbf{v}$ is written $v_1$. In numpy, how do you access that same number?
Answer
`v[0]`. Explanation: math is 1-indexed, numpy is 0-indexed, so $v_i$ corresponds to `v[i-1]` — a reliable source of off-by-one bugs.Q11. Why does this book treat "a vector is a list of numbers" as essential even when the list has 300 entries and can't be drawn?
Answer
Because the *list* view scales: the same componentwise add/scale that move 2D arrows also operate on 300-dimensional data (e.g., word embeddings), where no arrow can be drawn. Geometry gives intuition; coordinates give computational power.Q12. Can any real scalar turn $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$ into $\begin{bmatrix} 0 \\ 1 \end{bmatrix}$? Why or why not?