Vectors are the building blocks of geometry in motion. They pack a punch with both size and direction, letting us map out forces and movements in space. Think of them as arrows that point the way and tell us how far to go.

Adding and subtracting vectors is like playing connect-the-dots. We can join them tip-to-tail or use math to combine their parts. Multiplying by numbers stretches or shrinks vectors, giving us new tools to solve real-world problems.

Vector Fundamentals

Properties of geometric vectors

• Vectors possess both magnitude and direction
  • Magnitude signifies the length of the vector (distance between initial and terminal points)
  • Direction specifies the orientation of the vector in space (angle relative to a reference axis)
• Vectors are graphically depicted as arrows in geometric contexts
  • Arrow tail marks the initial point (starting position)
  • Arrow head indicates the terminal point (ending position)
• Vectors are considered equivalent when they have identical magnitudes and directions, irrespective of their spatial locations
  • Translating a vector parallel to itself does not alter its properties
• Parallel vectors share the same direction but may differ in magnitude
  • Scaling a vector changes its length while maintaining its direction
• Zero vector has a magnitude of zero and lacks a specific direction
  • Represented by a point rather than an arrow

Vector Operations

Vector addition and subtraction

• Graphical vector addition
  • Place the tail of the second vector at the head of the first vector to add them graphically
  • Resultant vector extends from the tail of the first vector to the head of the second vector (triangle law of vector addition)
• Algebraic vector addition
  • Express vectors as ordered pairs $(a, b)$ in a 2D coordinate system
  • Add corresponding components to perform algebraic vector addition: $(a_1, b_1) + (a_2, b_2) = (a_1 + a_2, b_1 + b_2)$
• Graphical vector subtraction
  • Reverse the direction of the vector being subtracted and add it to the other vector for graphical subtraction
  • Resultant vector extends from the tail of the first vector to the head of the reversed second vector
• Algebraic vector subtraction
  • Subtract corresponding components to perform algebraic vector subtraction: $(a_1, b_1) - (a_2, b_2) = (a_1 - a_2, b_1 - b_2)$

Scalar multiplication of vectors

• Scalar multiplication involves multiplying a vector by a real number (scalar) to produce a new vector
  • Multiplying vector $\vec{v}$ by scalar $c$ yields a new vector $c\vec{v}$
• Scalar multiplication alters the magnitude of the vector
  • $|c| > 1$ results in a longer vector than the original
  • $0 < |c| < 1$ results in a shorter vector than the original
• Scalar multiplication can modify the direction of the vector
  • $c > 0$ preserves the original vector's direction
  • $c < 0$ reverses the original vector's direction
• Scalar multiplication by zero yields the zero vector

Magnitude and direction in plane

• The magnitude of a vector $\vec{v} = (a, b)$, denoted as $|\vec{v}|$, is calculated using the Pythagorean theorem: $|\vec{v}| = \sqrt{a^2 + b^2}$
  • Magnitude represents the length of the vector
• The direction of a vector is described by the angle it forms with the positive x-axis
  • Angle $\theta$ is calculated using the arctangent function: $\theta = \tan^{-1}(\frac{b}{a})$, where $a$ and $b$ are the vector's components
• Unit vectors have a magnitude of 1 and solely represent direction
  • To find the unit vector in the same direction as $\vec{v}$, divide the vector by its magnitude: $\hat{v} = \frac{\vec{v}}{|\vec{v}|}$
  • Unit vectors are useful for describing directions without considering magnitude