10.8 Vectors

Vectors are the superheroes of mathematics, swooping in to tackle problems involving direction and magnitude. They're not just lines with arrows; they're powerful tools for representing everything from forces in physics to data in computer graphics.

Understanding vectors is like unlocking a secret language of motion and space. We'll explore how to add, multiply, and manipulate these mathematical marvels, seeing how they simplify complex problems and bring abstract concepts to life in the real world.

Vector Fundamentals

Vectors as geometric objects

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• Represented as directed line segments
  • Length represents vector's magnitude (size or length)
  • Arrow indicates vector's direction (orientation in space)
• Denoted using boldface letters ($\mathbf{v}$) or letters with arrows ($\vec{v}$)
• Magnitude of vector $\vec{v}$ denoted as $|\vec{v}|$ (absolute value or norm)
• Two vectors equal if same magnitude and direction (regardless of position)

Vector magnitude and direction

• Magnitude of vector $\vec{v} = (x, y)$ in plane given by formula:
  • $|\vec{v}| = \sqrt{x^2 + y^2}$ (Pythagorean theorem)
• Direction described using:
  • Angle measures (45° counterclockwise from positive x-axis)
  • Cardinal directions (northeast, southwest)
• Direction of vector $\vec{v} = (x, y)$ found using arctangent function:
  • $\theta = \tan^{-1}(\frac{y}{x})$, where $\theta$ is angle with positive x-axis
  • Determines quadrant vector points to (I, II, III, or IV)

Basic vector operations

• Vector addition: To add vectors $\vec{u} = (u_1, u_2)$ and $\vec{v} = (v_1, v_2)$, add corresponding components:
  • $\vec{u} + \vec{v} = (u_1 + v_1, u_2 + v_2)$ (component-wise addition)
  • Visualized using parallelogram rule or triangle rule (head-to-tail method)
• Scalar multiplication: To multiply vector $\vec{v} = (x, y)$ by scalar $c$, multiply each component by $c$:
  • $c\vec{v} = (cx, cy)$ (distributes to each component)
  • Changes magnitude but not direction (unless $c$ negative)

Vector Representation and Products

Vectors in component form

• Standard unit vectors in plane:
  • $\hat{i} = (1, 0)$, points in positive x-direction (horizontal)
  • $\hat{j} = (0, 1)$, points in positive y-direction (vertical)
• Vector $\vec{v} = (x, y)$ expressed in component form as:
  • $\vec{v} = x\hat{i} + y\hat{j}$ (linear combination of unit vectors)
• Allows for easy vector addition and scalar multiplication (algebraic operations)

Dot product of vectors

• Dot product of vectors $\vec{u} = (u_1, u_2)$ and $\vec{v} = (v_1, v_2)$ defined as:
  • $\vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2$ (sum of products of corresponding components)
• Results in scalar value (not a vector)
• Related to angle $\theta$ between vectors:
  • $\vec{u} \cdot \vec{v} = |\vec{u}||\vec{v}|\cos(\theta)$ (magnitude and direction)
• Used to determine:
  • Orthogonality (perpendicularity): $\vec{u} \cdot \vec{v} = 0$ (90° angle)
  • Projection of one vector onto another (component in direction of other vector)

Unit vectors in direction

• Unit vector has magnitude of 1 (length of 1)
• To construct unit vector $\hat{u}$ in direction of vector $\vec{v} = (x, y)$:
  • Divide $\vec{v}$ by its magnitude: $\hat{u} = \frac{\vec{v}}{|\vec{v}|} = (\frac{x}{\sqrt{x^2 + y^2}}, \frac{y}{\sqrt{x^2 + y^2}})$ (normalization)
• Useful for representing directions without considering magnitude (pure direction)

Applications of Vectors

Vector applications in physics

• Represent physical quantities with magnitude and direction:
  • Displacement, velocity, acceleration (kinematics)
  • Force, momentum (dynamics)
• Analyze and solve problems involving:
  1. Find resultant force acting on object
  2. Express each force vector in component form
  3. Add force vectors component-wise to find resultant force vector
  4. Calculate magnitude and direction of resultant force vector
• Motion in 2D or 3D (projectile motion, circular motion)
• Work done by force along displacement (dot product)
• Vector fields describe physical quantities that vary in space (e.g., electromagnetic fields)

Advanced Vector Concepts

Vector spaces and calculus

• Vector space: A set of vectors that can be added and scaled, following specific axioms
• Linear combination: Expressing a vector as a sum of scaled vectors from a given set
• Vector calculus: Branch of mathematics dealing with differentiation and integration of vector fields
• Coordinate systems: Frameworks for specifying vector positions (e.g., Cartesian, polar, spherical)