Vectors are the mathematical superheroes of motion and force. They pack both size and direction into one neat package, letting us describe everything from a gentle breeze to a rocket's trajectory. Understanding vectors is key to mastering physics and engineering concepts.

Vector operations are like building blocks for solving complex problems. By adding, subtracting, and multiplying vectors, we can break down tricky scenarios into manageable parts. This toolbox of techniques opens doors to tackling real-world challenges in fields from robotics to video game design.

Vector Fundamentals

Geometric and algebraic vector interpretation

• Vectors possess both magnitude and direction
  • Geometrically depicted as directed line segments with an initial and terminal point (displacement, force)
  • Algebraically represented using ordered pairs $\langle a, b \rangle$, component form $a\hat{i} + b\hat{j}$, or unit vector notation
• Vectors with identical magnitude and direction are equal, irrespective of their initial points (equivalent forces)

Vector magnitude and direction

• The magnitude of a vector $\vec{v} = \langle a, b \rangle$ is calculated as $|\vec{v}| = \sqrt{a^2 + b^2}$ (distance formula)
• A vector's direction is described by the angle it forms with the positive x-axis
  • The angle $\theta$ is computed using $\tan \theta = \frac{b}{a}$, where $a$ and $b$ are the vector's components (slope of the line)

Vector operations

• Vector addition: To add vectors $\vec{u} = \langle a_1, b_1 \rangle$ and $\vec{v} = \langle a_2, b_2 \rangle$, add their corresponding components $\vec{u} + \vec{v} = \langle a_1 + a_2, b_1 + b_2 \rangle$ (tip-to-tail method)
  • The parallelogram law provides a geometric interpretation of vector addition
• Vector subtraction: To subtract vector $\vec{v}$ from $\vec{u}$, subtract their corresponding components $\vec{u} - \vec{v} = \langle a_1 - a_2, b_1 - b_2 \rangle$ (subtracting displacements)
• Scalar multiplication: To multiply a vector $\vec{v} = \langle a, b \rangle$ by a scalar $c$, multiply each component by the scalar $c\vec{v} = \langle ca, cb \rangle$ (scaling a force)

Vector Representations and Operations

Vector forms and unit vectors

• Component form: A vector $\vec{v}$ is written as $\vec{v} = \langle a, b \rangle$, where $a$ and $b$ are the horizontal and vertical components (coordinates)
• Unit vector notation: A vector $\vec{v}$ is expressed as a linear combination of standard unit vectors $\hat{i}$ (horizontal) and $\hat{j}$ (vertical) $\vec{v} = a\hat{i} + b\hat{j}$ (basis vectors)

I and j notation for vectors

• Addition: $\vec{u} + \vec{v} = (a_1\hat{i} + b_1\hat{j}) + (a_2\hat{i} + b_2\hat{j}) = (a_1 + a_2)\hat{i} + (b_1 + b_2)\hat{j}$
• Subtraction: $\vec{u} - \vec{v} = (a_1\hat{i} + b_1\hat{j}) - (a_2\hat{i} + b_2\hat{j}) = (a_1 - a_2)\hat{i} + (b_1 - b_2)\hat{j}$
• Scalar multiplication: $c\vec{v} = c(a\hat{i} + b\hat{j}) = (ca)\hat{i} + (cb)\hat{j}$

Dot products of vectors

• The dot product of vectors $\vec{u} = \langle a_1, b_1 \rangle$ and $\vec{v} = \langle a_2, b_2 \rangle$ is a scalar given by $\vec{u} \cdot \vec{v} = a_1a_2 + b_1b_2$ (sum of component products)
• The dot product finds the angle $\theta$ between two vectors $\vec{u} \cdot \vec{v} = |\vec{u}||\vec{v}|\cos \theta$ (projection formula)
• Dot products determine orthogonality (perpendicular if $\vec{u} \cdot \vec{v} = 0$) and work done by a force $W = \vec{F} \cdot \vec{d}$
• Vector projection can be calculated using the dot product formula

Advanced Vector Operations

• Cross product: A binary operation on two vectors in three-dimensional space, resulting in a vector perpendicular to both input vectors
• Linear combination: Expressing a vector as a sum of scalar multiples of other vectors
• Vector space: A collection of vectors that is closed under vector addition and scalar multiplication

Vector applications in real-world problems

• Vectors model physical quantities like displacement, velocity, acceleration, and force (motion, mechanics)
• Real-world applications span navigation, physics, engineering, and computer graphics (GPS, video games)
• Problem-solving involves identifying relevant vector quantities, representing them mathematically, and performing vector operations to obtain the desired result (projectile motion, equilibrium)