# Vectors in 2D and 3D

Physics I · Linear Algebra and Geometry · https://tryals.app/learn/physics-i/vectors-in-2d-and-3d

A **vector** is a quantity possessing both **magnitude** and **direction**, applied in physics for velocity, force, and displacement.

Expressed via standard basis vectors $\hat{i}$, $\hat{j}$, $\hat{k}$:

$\vec{v} = v_x\hat{i} + v_y\hat{j} + v_z\hat{k}$

| Operation | Formula | Meaning |
| --- | --- | --- |
| **Addition** | $\vec{u}+\vec{v}$ | $(u_x+v_x, u_y+v_y, u_z+v_z)$ |
| **Scalar mult.** | $\alpha\vec{v}$ | Scaling by $\alpha$ |
| **Magnitude** | $\|\vec{v}\|$ | $\sqrt{v_x^2 + v_y^2 + v_z^2}$ |
| **Unit vector** | $\hat{v}$ | $\vec{v}\,/\,\|\vec{v}\|$ |

**Common pitfall:** Vectors add tip-to-tail, not by adding scalar lengths. Walking 3 km east then 4 km north yields a 5 km displacement, not 7 km. Magnitudes only add when directions align.

## Vector Properties

Vector operations obey strict algebraic laws essential for multidimensional problem solving.

| Property | Algebraic Rule |
| --- | --- |
| **Commutativity** | $\vec{u}+\vec{v} = \vec{v}+\vec{u}$ |
| **Associativity** | $(\vec{u}+\vec{v})+\vec{w} = \vec{u}+(\vec{v}+\vec{w})$ |
| **Zero vector** | $\vec{v}+\vec{0} = \vec{v}$ |

**Worked Example:** For $\vec{v} = (3, 4, 0)$, magnitude is $\|\vec{v}\| = \sqrt{3^2 + 4^2 + 0^2} = 5$. Its unit vector is $\hat{v} = (3/5, 4/5, 0)$.

**Tip:** Always sketch vectors before computing. A quick diagram catches sign errors that raw algebra frequently misses.

## Practice questions

5 of this lesson's 11 practice questions, with answers. The full set is in the app.

### 1. Sort each expression by what kind of object it produces ($u, v$ are vectors).

**Answer:**

- Vector: $u + v$, $3u$, $u \times v$ (cross product, in 3D)
- Scalar: $u \cdot v$ (dot product), $\|u\|$ (norm)

**Why:** Type-checking expressions, vector or scalar?, catches most linear-algebra errors before any arithmetic. Adding a scalar to a vector, for example, is as meaningless as adding seconds to metres.

Page: https://tryals.app/practice/physics-i/vectors-in-2d-and-3d/sort-each-expression-by-what-kind-of-object-it-produces-u-v-are

### 2. Which properties hold for vector addition in ℝⁿ?

A. Every vector has a multiplicative inverse
B. Associative: (u+v)+$w = u+(v+w)$
C. Commutative: u + $v = v + u$
D. There exists an additive identity (zero vector)

**Answer:** B. Associative: (u+v)+$w = u+(v+w)$; C. Commutative: u + $v = v + u$; D. There exists an additive identity (zero vector)

**Why:** Vectors do not have multiplicative inverses; that concept applies to scalars.

Page: https://tryals.app/practice/physics-i/vectors-in-2d-and-3d/which-properties-hold-for-vector-addition-in

### 3. Sort these into 2D or 3D vector concepts.

**Answer:**

- 2D only: Rotation by angle $\theta$ in the plane, Signed area via determinant |u v|
- Both 2D and 3D: Angle between two vectors from $\cos\theta$, Dot product u $\cdot v$, Unit vector

**Why:** Most vector operations generalise to any dimension; some are dimension-specific.

Page: https://tryals.app/practice/physics-i/vectors-in-2d-and-3d/sort-these-into-2d-or-3d-vector-concepts

### 4. A hiker walks 3 km east, then 4 km north. Their GPS "displacement" reads 5 km, but their pedometer reads 7 km. Which statement explains the disagreement?

A. The pedometer double-counts steps on turns
B. Both should read 7 km; the GPS malfunctioned
C. Displacement is a vector sum, not a total
D. The GPS is inaccurate over long walks

**Answer:** C. Displacement is a vector sum, not a total

**Why:** Vectors add geometrically, not arithmetically: perpendicular legs of 3 and 4 compose into a displacement of 5. Distance walked (a scalar) and displacement achieved (a vector) answer different questions, the core distinction of vector thinking.

Page: https://tryals.app/practice/physics-i/vectors-in-2d-and-3d/a-hiker-walks-3-km-east-then-4-km-north-their-gps-displacement

### 5. The dot product gives a scalar and the cross product gives a vector. Why does that difference match what each is used for in physics?

A. Because scalars are easier to compute than vectors
B. Because work has no direction and torque has an axis
C. Because the two products measure the same thing in different notations
D. Because the dot product is defined in all dimensions and the cross product only in 3D

**Answer:** B. Because work has no direction and torque has an axis

**Why:** The mathematics is chosen to match the physics. A quantity with an axis needs a vector output; a quantity that is just an amount needs a scalar.

Page: https://tryals.app/practice/physics-i/vectors-in-2d-and-3d/the-dot-product-gives-a-scalar-and-the-cross-product-gives-a-vector