# Vectors in the Plane and Space

Mathematics I · Matrices and Vectors · https://tryals.app/learn/mathematics-i/vectors-in-the-plane-and-space

## Vectors and Basic Operations

A **vector** is a quantity with magnitude and direction, represented by an arrow. In coordinates, a 2D vector is $(v_1, v_2)$ and a 3D vector is $(v_1, v_2, v_3)$; generally $\mathbb{R}^n$ uses $n$ components. **Components** are signed displacements along axes.

Two operations govern vectors:
| Operation | Algebraic Rule | Geometric Rule |
| --- | --- | --- |
| **Addition** | $(a_1, a_2) + (b_1, b_2) = (a_1+b_1, a_2+b_2)$ | **Tip-to-tail** placement; sum runs from first tail to last tip. |
| **Scalar Multiplication** | $c(v_1, v_2) = (cv_1, cv_2)$ | Scales length by $|c|$; negative $c$ reverses direction. |

## Magnitude and Normalization

The **magnitude** (length or **norm**) is calculated via the Pythagorean theorem: $\|v\| = \sqrt{v_1^2 + v_2^2}$ in the plane, extending to space as $\|v\| = \sqrt{v_1^2 + v_2^2 + v_3^2}$. A **unit vector** has length 1. **Normalizing** a nonzero vector divides it by its length: $\frac{v}{\|v\|}$.

**Common pitfall:** Never add vectors by adding their **magnitudes**, and never confuse magnitude with components. Vectors add **componentwise**. The triangle inequality states $\|u + v\| \le \|u\| + \|v\|$, with equality holding only when $u$ and $v$ point in the exact same direction.

## Practice questions

3 of this lesson's 9 practice questions, with answers. The full set is in the app.

### 1. Two arrows of the same length represent the same vector.

**Answer:** False

**Why:** False. A vector is direction *and* magnitude, so two arrows of equal length pointing differently are different vectors. Position is what does not matter: the arrow from (0,0) to (3,4) and the one from (1,1) to (4,5) are the same vector, which is what lets them slide for tip-to-tail addition.

Page: https://tryals.app/practice/mathematics-i/vectors-in-the-plane-and-space/two-arrows-of-the-same-length-represent-the-same-vector

### 2. A pilot flies 100 km north then 100 km east, and a second flies 100 km east then 100 km north. They land in the same place. Why does vector addition commute when matrix multiplication does not?

A. Because vectors are simpler objects than matrices and obey more rules
B. Because addition combines displacements while matrices compose actions
C. Because the two flights are not actually equivalent, only approximately so
D. Because commutativity holds in two dimensions but fails in three

**Answer:** B. Because addition combines displacements while matrices compose actions

**Why:** Displacements add up to the same corner whichever leg you walk first. A matrix does something *to* space, and rotate-then-reflect is a different operation from reflect-then-rotate, which is why the same subject has one commutative operation and one that is not, and the difference is quantity against action.

Page: https://tryals.app/practice/mathematics-i/vectors-in-the-plane-and-space/a-pilot-flies-100-km-north-then-100-km-east-and-a-second-flies-100

### 3. Two vectors are parallel when one is a scalar multiple of the other. Sort each pair.

**Answer:**

- Parallel: (1, 2) and (3, 6), (2, -3) and (-4, 6), (0, 5) and (0, -2)
- Not parallel: (1, 2) and (2, 1), (1, 1) and (1, -1), (3, 6) and (1, 3)

**Why:** (3, 6) = 3(1, 2) and (-4, 6) = -2(2, -3), so both pairs are parallel; opposite directions still count. (0, 5) and (0, -2) both lie along the vertical axis. But (3, 6) and (1, 3) fail: the first components need a factor of 3 and the seconds a factor of 2, and swapping components as in (1, 2) and (2, 1) is never a scaling.

Page: https://tryals.app/practice/mathematics-i/vectors-in-the-plane-and-space/two-vectors-are-parallel-when-one-is-a-scalar-multiple-of-the-other