Introduction to Classical Physics I Lecture 2 Vectors Vector Addition

Introduction to Vectors and Vector Addition

  • This set of notes covers a review of vectors and key operations used in classical physics:
    • Properties of vectors
    • Vector addition and subtraction
    • Resolving a vector into components
    • Describing the resultant vector from addition using graphical and analytical methods
    • Basic vector multiplication concepts (scalar multiplication, dot product, cross product) as context
  • Author/Course context: Prof. James R. Zabel, Iowa State University; Vectors & Vector Addition for PHYS-2310 (Fall 2025).

Scalar vs Vector; Definitions and Representations

  • Scalar vs Vector definitions:
    • Scalar: quantity with magnitude only (no direction). Examples: mass, length, time, temperature, volume, etc.
    • Example values: 85 kg, 175 cm, 437 ns, 37 °C, 1.9 m^3, etc.
    • Vector: quantity with both magnitude and direction; often represented in bold or with an arrow (e.g., A or A).
  • Vector representations:
    • Cartesian representation:
    • (#\mathbf{A}) = (Ax, Ay, A_z)
    • Basis representation:
    • (#\mathbf{A}) = Ax \hat{i} + Ay \hat{j} + A_z \hat{k}
    • Polar/2D representation:
    • (#\mathbf{A}) = |\mathbf{A}| ⟨θ⟩, where |\mathbf{A}| is the magnitude and θ is the angle measured counterclockwise from the +x axis.
    • The vector is the line from the origin to its head when drawn in these representations.
  • Important note: Vectors are defined by their magnitude and direction, not by their location in space.

Right Triangles, Pythagorean Theorem, and Trigonometry Review

  • Right triangle definition:
    • One angle is 90°, hypotenuse is the side opposite the right angle.
  • Pythagorean theorem:
    • C2=A2+B2C^2 = A^2 + B^2
    • Here, C is the hypotenuse, A and B are the other two sides.
  • Trigonometric identities (SOH-CAH-TOA):
    • sin⁡α=oppositehypotenuse=OH\sin \alpha = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{O}{H}
    • cos⁡α=adjacenthypotenuse=AH\cos \alpha = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{A}{H}
    • tan⁡α=oppositeadjacent=OA\tan \alpha = \frac{\text{opposite}}{\text{adjacent}} = \frac{O}{A}
  • These identities underpin resolving vectors into components.

Vector Addition and Subtraction

  • Vector addition (head-to-tail rule):
    • To add (#\mathbf{A}) and (#\mathbf{B}), plot (#\mathbf{A}) first, then place the tail of (#\mathbf{B}) at the head of (#\mathbf{A}).
    • The resultant vector