Physics isn't just about numbers; it's about describing the world around us. Vectors and scalars are the building blocks we use to understand motion, forces, and energy. They help us break down complex problems into manageable parts.

Vectors pack a punch with both size and direction, while scalars keep it simple with just magnitude. Mastering vector operations and analysis is key to tackling real-world physics problems, from projectile motion to electromagnetic fields.

Vector and Scalar Quantities

Scalar vs vector quantities

• Scalar quantities described by magnitude alone (mass, temperature, time, speed, energy)
• Vector quantities described by both magnitude and direction (displacement, velocity, acceleration, force, momentum)

Vector operations and calculations

• Vector addition combines vectors graphically (tip-to-tail) or analytically (component-wise) resulting in $\vec{R} = \vec{A} + \vec{B}$
• Vector subtraction adds negative of vector $\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$
• Scalar multiplication changes vector magnitude, possibly direction $c\vec{A} = (cA_x, cA_y, cA_z)$

Vector magnitude and direction

• Two-dimensional vector magnitude $|\vec{A}| = \sqrt{A_x^2 + A_y^2}$
• Three-dimensional vector magnitude $|\vec{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2}$
• Direction angle $\theta = \tan^{-1}(\frac{A_y}{A_x})$ considering quadrants
• Unit vectors have magnitude of 1, calculated as $\hat{A} = \frac{\vec{A}}{|\vec{A}|}$

Vector components and analysis

• Vector components: x-component $A_x = A \cos\theta$, y-component $A_y = A \sin\theta$
• Vector resolution breaks down vector into x and y components for problem-solving
• Applies to physical situations (projectile motion, forces on inclined planes, relative motion)
• Vector dot product yields scalar result $\vec{A} \cdot \vec{B} = AB \cos\theta$ used in work and energy calculations
• Vector cross product yields vector result $|\vec{A} \times \vec{B}| = AB \sin\theta$ with direction from right-hand rule, used in torque and angular momentum