Two-Dimensional Vectors & Force Resolution Flashcards

Question 1

What is the difference between a scalar and a vector in physics?

Accepted Answer

- **Scalar**: A physical quantity specified solely by magnitude and unit (e.g. mass, temperature, time, energy).
- **Vector**: A quantity possessing both **magnitude** and **direction** (e.g. displacement, velocity, force, acceleration).

Question 2

How do you calculate the magnitude of a 2D vector $\mathbf{F} = [F_x, F_y]$?

Accepted Answer

$$\|\mathbf{F}\| = \sqrt{F_x^2 + F_y^2}$$
Derived directly from the Pythagorean theorem on Cartesian coordinates.

Question 3

Given magnitude $F$ and angle $	heta$ relative to the positive x-axis, what are $F_x$ and $F_y$?

Accepted Answer

$$F_x = F \cos	heta \quad 	ext{and} \quad F_y = F \sin	heta$$
Decomposing the force into orthogonal horizontal and vertical projections.

Question 4

State Newton\'s Second Law in two dimensions using Cartesian components.

Accepted Answer

$$\sum F_x = m a_x \quad 	ext{and} \quad \sum F_y = m a_y$$
The horizontal forces independently govern horizontal acceleration, while vertical forces govern vertical acceleration.

Question 5

On an inclined plane at angle $	heta$, how is gravitational force $F_g = mg$ decomposed?

Accepted Answer

- **Parallel to incline** (driving down slope): $$F_{\parallel} = mg \sin	heta$$
- **Perpendicular to incline** (pressing into surface): $$F_{\perp} = mg \cos	heta$$

Question 6

What is the Normal Force ($N$) on a mass $m$ resting on an incline of angle $	heta$?

Accepted Answer

$$N = mg \cos	heta$$
The surface pushes back perpendicularly against the mass to balance the perpendicular component of gravity.

Question 7

What is the formula for kinetic friction force $f_k$?

Accepted Answer

$$f_k = \mu_k N$$
Where $\mu_k$ is the coefficient of kinetic friction and $N$ is the normal force. Friction always opposes relative surface motion.

Question 8

How are two vectors $\mathbf{A}$ and $\mathbf{B}$ added analytically?

Accepted Answer

$$\mathbf{R} = \mathbf{A} + \mathbf{B} = [A_x + B_x, \; A_y + B_y]$$
Vector addition is performed component-wise.

Question 9

What are standard Cartesian unit vectors $\hat{i}$ and $\hat{j}$?

Accepted Answer

Dimensionless basis vectors of unit length pointing along coordinate axes:
$$\hat{i} = [1, 0] \quad 	ext{and} \quad \hat{j} = [0, 1]$$
Any 2D vector can be expressed as $\mathbf{v} = v_x \hat{i} + v_y \hat{j}$.

Question 10

What is the condition for static translational equilibrium in 2D?

Accepted Answer

The vector sum of all external forces must be zero:
$$\sum \mathbf{F} = \mathbf{0} \iff \sum F_x = 0 \quad 	ext{and} \quad \sum F_y = 0$$

Two-Dimensional Vectors & Force Resolution

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