2.1 Scalars and Vectors

Physics uses two types of quantities: scalars and vectors. Scalars have only magnitude, like temperature or time. Vectors have both magnitude and direction, like velocity or force. Understanding these is crucial for describing physical phenomena accurately.

Vectors can be added, subtracted, and multiplied by scalars. These operations are essential for solving physics problems involving multiple forces or displacements. Vector algebra helps us analyze complex systems and predict outcomes in mechanics and other areas of physics.

Introduction to Scalars and Vectors

Scalars vs vectors in physics

• Scalars quantify physical properties with a single value (magnitude) independent of direction
  • Temperature (℃), time (seconds), speed (m/s), volume (liters), energy (joules)
• Vectors quantify physical properties with both magnitude and direction
  • Represented by an arrow, length indicates magnitude, orientation indicates direction
  • Displacement (meters north), velocity (m/s east), acceleration (m/s² down), force (newtons up), momentum (kg⋅m/s southwest)

Components of vectors

• Magnitude quantifies the size or extent of the vector quantity
  • Represented by the length of the vector arrow
  • Always a positive scalar value
• Direction specifies the orientation of the vector quantity in space
  • Represented by the direction the vector arrow points
  • Described using cardinal directions (north, southwest) or angles relative to a reference axis (30° above the x-axis)
• Vector decomposition breaks a vector into its components along different axes
  • Useful for analyzing complex motions or forces in multiple dimensions

Vector Operations and Applications

Scalar multiplication of vectors

• Multiplying a vector by a scalar changes its magnitude without affecting its direction
  • Positive scalar multiplication lengthens the vector (2 × $\vec{v}$ doubles the length)
  • Negative scalar multiplication reverses the vector's direction and lengthens it (-1 × $\vec{v}$ flips and maintains length)
• The resulting vector is parallel to the original vector
• Scalar multiplication formula: $\vec{v} = c\vec{u}$
  • $\vec{v}$ is the resulting vector
  • $c$ is the scalar
  • $\vec{u}$ is the original vector

Vector addition and subtraction

• Vector addition combines vectors graphically or analytically
  1. Graphically, place the tail of one vector at the head of the other
  2. Draw the resultant vector from the tail of the first to the head of the second
  3. Analytically, add the corresponding components: $\vec{R} = \vec{A} + \vec{B} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j}$
• Vector subtraction finds the difference between vectors graphically or analytically
  1. Graphically, place the tails of the vectors together
  2. Draw the resultant vector from the head of the subtracted vector to the head of the other vector
  3. Analytically, subtract the corresponding components: $\vec{R} = \vec{A} - \vec{B} = (A_x - B_x)\hat{i} + (A_y - B_y)\hat{j}$

Vector algebra for physics problems

• Resultant force is the net force acting on an object due to multiple individual forces
  • Found by vector addition of all the individual forces: $\vec{F}_{net} = \vec{F}_1 + \vec{F}_2 + \ldots + \vec{F}_n$
  • A book on a table experiences the downward force of gravity and the upward normal force from the table
• Resultant displacement is the net change in position due to multiple individual displacements
  • Found by vector addition of all the individual displacements: $\vec{s}_{net} = \vec{s}_1 + \vec{s}_2 + \ldots + \vec{s}_n$
  • A hiker's path can be broken into displacements (legs) and added to find the overall displacement (distance and direction) from the starting point

Interpretation of vector equations

• Vector equations describe relationships between vector quantities
  • Newton's second law, $\vec{F} = m\vec{a}$, relates force (a vector) to mass (a scalar) and acceleration (a vector)
  • The direction of the force and acceleration vectors must match
• Scalar equations describe relationships between scalar quantities
  • The kinetic energy equation, $K = \frac{1}{2}mv^2$, relates kinetic energy (a scalar) to mass (a scalar) and speed (a scalar)
  • Speed is the scalar magnitude of the velocity vector
• Interpreting these equations requires understanding the physical meaning of each term and how they relate to the system being analyzed

Advanced Vector Operations

• Dot product (scalar product) of two vectors results in a scalar quantity
  • Used to calculate work done by a force or to find the angle between two vectors
• Cross product of two vectors results in a vector perpendicular to both original vectors
  • Used in calculating torque or angular momentum
• Vector fields represent vector quantities that vary with position in space
  • Examples include electric fields, magnetic fields, and fluid flow fields